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3.4.1: Graphs of Logarithmic Functions

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Graphing Logarithmic Functions

Your math homework assignment is to find out which quadrants the graph of the function f(x)=4ln(x+3) falls in. On the way home, your best friend tells you, "This is the easiest homework assignment ever! All logarithmic functions fall in Quadrants I and IV." You're not so sure, so you go home and graph the function as instructed. Your graph falls in Quadrant I as your friend thought, but instead of Quadrant IV, it also falls in Quadrants II and III. Which one of you is correct?

Now that we are more comfortable with using these functions as inverses, let’s use this idea to graph a logarithmic function. Recall that functions are inverses of each other when they are mirror images over the line y=x. Therefore, if we reflect y=b x over y=x, then we will get the graph of y=log b x.

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Recall that an exponential function has a horizontal asymptote. Because the logarithm is its inverse, it will have a vertical asymptote. The general form of a logarithmic function is f(x)=a log b (x−h)+k and the vertical asymptote is x=h. The domain is x>h and the range is all real numbers. Lastly, if b>1, the graph moves up to the right. If 0<b<1, the graph moves down to the right.

Let's graph y=log 3 (x−4) and state the domain and range .

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To graph a logarithmic function without a calculator, start by drawing the vertical asymptote, at x=4. We know the graph is going to have the general shape of the first function above. Plot a few points, such as (5, 0), (7, 1), and (13, 2) and connect.

The domain is x>4 and the range is all real numbers.

Now, let's determine if (16, 1) is on y=log(x−6).

Plug in the point to the equation to see if it holds true.

\(\ \begin{array}{l} 1=\log (16-6) \\ 1=\log 10 \\ 1=1 \end{array}\)

Yes, this is true, so (16, 1) is on the graph.

Finally, let's graph f(x)=2ln(x+1).

To graph a natural log, we can use a graphing calculator. Press Y= and enter in the function, Y=2ln(x+1), GRAPH .

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Earlier, you were asked to determine if your friend was correct.

The vertical asymptote of the function f(x)=4ln(x+3) is x=−3. Since x will approach −3 but never quite reach it, x can assume some negative values. Hence, the function will fall in Quadrants II and III. Therefore, you are correct and your friend is wrong.

Graph \(\ y=\log _{\frac{1}{4}} x+2\) in an appropriate window.

First, there is a vertical asymptote at x=0. Now, determine a few easy points, points where the log is easy to find; such as (1, 2), (4, 1), (8, 0.5), and (16, 0).

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To graph a logarithmic function using a TI-83/84, enter the function into Y= and use the Change of Base Formula: \(\ \log _{a} x=\frac{\log _{b} x}{\log _{b} a}\). The keystrokes would be: \(\ Y=\frac{\log (x)}{\log \left(\frac{1}{4}\right)}+2\), GRAPH

To see a table of values, press 2 nd → GRAPH .

Graph y=−logx using a graphing calculator. Find the domain and range .

The keystrokes are Y=−log(x), GRAPH .

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The domain is x>0 and the range is all real numbers.

Is (-2, 1) on the graph of \(\ f(x)=\log _{\frac{1}{2}}(x+4)\)?

Plug (-2, 1) into \(\ f(x)=\log _{\frac{1}{2}}(x+4)\) to see if the equation holds true.

\(\ \begin{array}{l} 1=\log _{\frac{1}{2}}(-2+4) \\ 1=\log _{\frac{1}{2}} 2 \\ 1 \neq-1 \end{array}\)

Therefore, (-2, 1) is not on the graph. However, (-2, -1) is.

Graph the following logarithmic functions without using a calculator. State the equation of the asymptote, the domain and the range of each function.

  • \(\ y=\log _{5} x\)
  • \(\ y=\log _{2}(x+1)\)
  • \(\ y=\log (x)-4\)
  • \(\ y=\log _{\frac{1}{3}}(x-1)+3\)
  • \(\ y=-\log _{\frac{1}{2}}(x+3)-5\)
  • \(\ y=\log _4(2-x)+2\)

Graph the following logarithmic functions using your graphing calculator.

  • \(\ y=\ln(x+6)-1\)
  • \(\ y=-\ln(x-1)+2\)
  • \(\ y=\log(1-x)+3\)
  • \(\ y=\log(x+2)-4\)
  • How would you graph \(\ y=\log_4x\) on the graphing calculator? Graph the function.
  • Graph \(\ y=\log_{\frac{3}{4}}x\) on the graphing calculator.
  • Is (3, 8) on the graph of \(\ y=\log_3(2x-3)+7\)?
  • Is (9, -2) on the graph of \(\ y=\log_{\frac{1}{4}}(x-5)\)?
  • Is (4, 5) on the graph of \(\ y=5\log_2(8-x)\)?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.7.

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  • 6.3 Logarithmic Functions
  • Introduction to Prerequisites
  • 1.1 Real Numbers: Algebra Essentials
  • 1.2 Exponents and Scientific Notation
  • 1.3 Radicals and Rational Exponents
  • 1.4 Polynomials
  • 1.5 Factoring Polynomials
  • 1.6 Rational Expressions
  • Key Equations
  • Key Concepts
  • Review Exercises
  • Practice Test
  • Introduction to Equations and Inequalities
  • 2.1 The Rectangular Coordinate Systems and Graphs
  • 2.2 Linear Equations in One Variable
  • 2.3 Models and Applications
  • 2.4 Complex Numbers
  • 2.5 Quadratic Equations
  • 2.6 Other Types of Equations
  • 2.7 Linear Inequalities and Absolute Value Inequalities
  • Introduction to Functions
  • 3.1 Functions and Function Notation
  • 3.2 Domain and Range
  • 3.3 Rates of Change and Behavior of Graphs
  • 3.4 Composition of Functions
  • 3.5 Transformation of Functions
  • 3.6 Absolute Value Functions
  • 3.7 Inverse Functions
  • Introduction to Linear Functions
  • 4.1 Linear Functions
  • 4.2 Modeling with Linear Functions
  • 4.3 Fitting Linear Models to Data
  • Introduction to Polynomial and Rational Functions
  • 5.1 Quadratic Functions
  • 5.2 Power Functions and Polynomial Functions
  • 5.3 Graphs of Polynomial Functions
  • 5.4 Dividing Polynomials
  • 5.5 Zeros of Polynomial Functions
  • 5.6 Rational Functions
  • 5.7 Inverses and Radical Functions
  • 5.8 Modeling Using Variation
  • Introduction to Exponential and Logarithmic Functions
  • 6.1 Exponential Functions
  • 6.2 Graphs of Exponential Functions
  • 6.4 Graphs of Logarithmic Functions
  • 6.5 Logarithmic Properties
  • 6.6 Exponential and Logarithmic Equations
  • 6.7 Exponential and Logarithmic Models
  • 6.8 Fitting Exponential Models to Data
  • Introduction to Systems of Equations and Inequalities
  • 7.1 Systems of Linear Equations: Two Variables
  • 7.2 Systems of Linear Equations: Three Variables
  • 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
  • 7.4 Partial Fractions
  • 7.5 Matrices and Matrix Operations
  • 7.6 Solving Systems with Gaussian Elimination
  • 7.7 Solving Systems with Inverses
  • 7.8 Solving Systems with Cramer's Rule
  • Introduction to Analytic Geometry
  • 8.1 The Ellipse
  • 8.2 The Hyperbola
  • 8.3 The Parabola
  • 8.4 Rotation of Axes
  • 8.5 Conic Sections in Polar Coordinates
  • Introduction to Sequences, Probability and Counting Theory
  • 9.1 Sequences and Their Notations
  • 9.2 Arithmetic Sequences
  • 9.3 Geometric Sequences
  • 9.4 Series and Their Notations
  • 9.5 Counting Principles
  • 9.6 Binomial Theorem
  • 9.7 Probability

Learning Objectives

In this section, you will:

  • Convert from logarithmic to exponential form.
  • Convert from exponential to logarithmic form.
  • Evaluate logarithms.
  • Use common logarithms.
  • Use natural logarithms.

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes 4 . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, 5 like those shown in Figure 1 . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale 6 whereas the Japanese earthquake registered a 9.0. 7

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is 10 8 − 4 = 10 4 = 10,000 10 8 − 4 = 10 4 = 10,000 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is 10 x = 500 , 10 x = 500 , where x x represents the difference in magnitudes on the Richter Scale . How would we solve for x ? x ?

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve 10 x = 500. 10 x = 500. We know that 10 2 = 100 10 2 = 100 and 10 3 = 1000 , 10 3 = 1000 , so it is clear that x x must be some value between 2 and 3, since y = 10 x y = 10 x is increasing. We can examine a graph, as in Figure 2 , to better estimate the solution.

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function y = b x y = b x is one-to-one , so its inverse, x = b y x = b y is also a function. As is the case with all inverse functions, we simply interchange x x and y y and solve for y y to find the inverse function. To represent y y as a function of x , x , we use a logarithmic function of the form y = log b ( x ) . y = log b ( x ) . The base b b logarithm of a number is the exponent by which we must raise b b to get that number.

We read a logarithmic expression as, “The logarithm with base b b of x x is equal to y , y , ” or, simplified, “log base b b of x x is y . y . ” We can also say, “ b b raised to the power of y y is x , x , ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since 2 5 = 32 , 2 5 = 32 , we can write log 2 32 = 5. log 2 32 = 5. We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

Note that the base b b is always positive.

Because logarithm is a function, it is most correctly written as log b ( x ) , log b ( x ) , using parentheses to denote function evaluation, just as we would with f ( x ) . f ( x ) . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as log b x . log b x . Note that many calculators require parentheses around the x . x .

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = log b ( x ) y = log b ( x ) and y = b x y = b x are inverse functions.

Definition of the Logarithmic Function

A logarithm base b b of a positive number x x satisfies the following definition.

For x > 0 , b > 0 , b ≠ 1 , x > 0 , b > 0 , b ≠ 1 ,

  • we read log b ( x ) log b ( x ) as, “the logarithm with base b b of x x ” or the “log base b b of x . " x . "
  • the logarithm y y is the exponent to which b b must be raised to get x . x .

Also, since the logarithmic and exponential functions switch the x x and y y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b   is   ( 0 , ∞ ) . b   is   ( 0 , ∞ ) .
  • the range of the logarithm function with base b   is   ( − ∞ , ∞ ) . b   is   ( − ∞ , ∞ ) .

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

Given an equation in logarithmic form log b ( x ) = y , log b ( x ) = y , convert it to exponential form.

  • Examine the equation y = log b ( x ) y = log b ( x ) and identify b , y , and x . b , y , and x .
  • Rewrite log b ( x ) = y log b ( x ) = y as b y = x . b y = x .

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

  • ⓐ log 6 ( 6 ) = 1 2 log 6 ( 6 ) = 1 2
  • ⓑ log 3 ( 9 ) = 2 log 3 ( 9 ) = 2

First, identify the values of b , y , and x . b , y , and x . Then, write the equation in the form b y = x . b y = x .

Here, b = 6 , y = 1 2 , and   x = 6. b = 6 , y = 1 2 , and   x = 6. Therefore, the equation log 6 ( 6 ) = 1 2 log 6 ( 6 ) = 1 2 is equivalent to 6 1 2 = 6 . 6 1 2 = 6 .

Here, b = 3 , y = 2 , and   x = 9. b = 3 , y = 2 , and   x = 9. Therefore, the equation log 3 ( 9 ) = 2 log 3 ( 9 ) = 2 is equivalent to 3 2 = 9. 3 2 = 9.

  • ⓐ log 10 ( 1, 000, 000 ) = 6 log 10 ( 1, 000, 000 ) = 6
  • ⓑ log 5 ( 25 ) = 2 log 5 ( 25 ) = 2

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b , b , exponent x , x , and output y . y . Then we write x = log b ( y ) . x = log b ( y ) .

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

  • 2 3 = 8 2 3 = 8
  • 5 2 = 25 5 2 = 25
  • 10 − 4 = 1 10,000 10 − 4 = 1 10,000

First, identify the values of b , y , and x . b , y , and x . Then, write the equation in the form x = log b ( y ) . x = log b ( y ) .

Here, b = 2 , b = 2 , x = 3 , x = 3 , and y = 8. y = 8. Therefore, the equation 2 3 = 8 2 3 = 8 is equivalent to log 2 ( 8 ) = 3. log 2 ( 8 ) = 3.

Here, b = 5 , b = 5 , x = 2 , x = 2 , and y = 25. y = 25. Therefore, the equation 5 2 = 25 5 2 = 25 is equivalent to log 5 ( 25 ) = 2. log 5 ( 25 ) = 2.

Here, b = 10 , b = 10 , x = − 4 , x = − 4 , and y = 1 10,000 . y = 1 10,000 . Therefore, the equation 10 − 4 = 1 10,000 10 − 4 = 1 10,000 is equivalent to log 10 ( 1 10,000 ) = − 4. log 10 ( 1 10,000 ) = − 4.

  • ⓐ 3 2 = 9 3 2 = 9
  • ⓑ 5 3 = 125 5 3 = 125
  • ⓒ 2 − 1 = 1 2 2 − 1 = 1 2

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log 2 8. log 2 8. We ask, “To what exponent must 2 2 be raised in order to get 8?” Because we already know 2 3 = 8 , 2 3 = 8 , it follows that log 2 8 = 3. log 2 8 = 3.

Now consider solving log 7 49 log 7 49 and log 3 27 log 3 27 mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know 7 2 = 49. 7 2 = 49. Therefore, log 7 49 = 2 log 7 49 = 2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know 3 3 = 27. 3 3 = 27. Therefore, log 3 27 = 3 log 3 27 = 3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log 2 3 4 9 log 2 3 4 9 mentally.

  • We ask, “To what exponent must 2 3 2 3 be raised in order to get 4 9 ? 4 9 ? ” We know 2 2 = 4 2 2 = 4 and 3 2 = 9 , 3 2 = 9 , so ( 2 3 ) 2 = 4 9 . ( 2 3 ) 2 = 4 9 . Therefore, log 2 3 ( 4 9 ) = 2. log 2 3 ( 4 9 ) = 2.

Given a logarithm of the form y = log b ( x ) , y = log b ( x ) , evaluate it mentally.

  • Rewrite the argument x x as a power of b : b : b y = x . b y = x .
  • Use previous knowledge of powers of b b identify y y by asking, “To what exponent should b b be raised in order to get x ? x ? ”

Solving Logarithms Mentally

Solve y = log 4 ( 64 ) y = log 4 ( 64 ) without using a calculator.

First we rewrite the logarithm in exponential form: 4 y = 64. 4 y = 64. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

Solve y = log 121 ( 11 ) y = log 121 ( 11 ) without using a calculator.

Evaluating the Logarithm of a Reciprocal

Evaluate y = log 3 ( 1 27 ) y = log 3 ( 1 27 ) without using a calculator.

First we rewrite the logarithm in exponential form: 3 y = 1 27 . 3 y = 1 27 . Next, we ask, “To what exponent must 3 be raised in order to get 1 27 ? 1 27 ? ”

We know 3 3 = 27 , 3 3 = 27 , but what must we do to get the reciprocal, 1 27 ? 1 27 ? Recall from working with exponents that b − a = 1 b a . b − a = 1 b a . We use this information to write

Therefore, log 3 ( 1 27 ) = − 3. log 3 ( 1 27 ) = − 3.

Evaluate y = log 2 ( 1 32 ) y = log 2 ( 1 32 ) without using a calculator.

Using Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression log ( x ) log ( x ) means log 10 ( x ) . log 10 ( x ) . We call a base-10 logarithm a common logarithm . Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of the Common Logarithm

A common logarithm is a logarithm with base 10. 10. We write log 10 ( x ) log 10 ( x ) simply as log ( x ) . log ( x ) . The common logarithm of a positive number x x satisfies the following definition.

For x > 0 , x > 0 ,

We read log ( x ) log ( x ) as, “the logarithm with base 10 10 of x x ” or “log base 10 of x . x . ”

The logarithm y y is the exponent to which 10 10 must be raised to get x . x .

Given a common logarithm of the form y = log ( x ) , y = log ( x ) , evaluate it mentally.

  • Rewrite the argument x x as a power of 10 : 10 : 10 y = x . 10 y = x .
  • Use previous knowledge of powers of 10 10 to identify y y by asking, “To what exponent must 10 10 be raised in order to get x ? x ? ”

Finding the Value of a Common Logarithm Mentally

Evaluate y = log ( 1000 ) y = log ( 1000 ) without using a calculator.

First we rewrite the logarithm in exponential form: 10 y = 1000. 10 y = 1000. Next, we ask, “To what exponent must 10 10 be raised in order to get 1000?” We know

Therefore, log ( 1000 ) = 3. log ( 1000 ) = 3.

Evaluate y = log ( 1, 000, 000 ) . y = log ( 1, 000, 000 ) .

Given a common logarithm with the form y = log ( x ) , y = log ( x ) , evaluate it using a calculator.

  • Press [LOG] .
  • Enter the value given for x , x , followed by [ ) ] .
  • Press [ENTER] .

Finding the Value of a Common Logarithm Using a Calculator

Evaluate y = log ( 321 ) y = log ( 321 ) to four decimal places using a calculator.

  • Enter 321 , followed by [ ) ] .

Rounding to four decimal places, log ( 321 ) ≈ 2.5065. log ( 321 ) ≈ 2.5065.

Note that 10 2 = 100 10 2 = 100 and that 10 3 = 1000. 10 3 = 1000. Since 321 is between 100 and 1000, we know that log ( 321 ) log ( 321 ) must be between log ( 100 ) log ( 100 ) and log ( 1000 ) . log ( 1000 ) . This gives us the following:

Evaluate y = log ( 123 ) y = log ( 123 ) to four decimal places using a calculator.

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10 x = 500 10 x = 500 represents this situation, where x x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

We begin by rewriting the exponential equation in logarithmic form.

Next we evaluate the logarithm using a calculator:

  • Enter 500 , 500 , followed by [ ) ] .
  • To the nearest thousandth, log ( 500 ) ≈ 2.699. log ( 500 ) ≈ 2.699.

The difference in magnitudes was about 2.699. 2.699.

The amount of energy released from one earthquake was 8,500 8,500 times greater than the amount of energy released from another. The equation 10 x = 8500 10 x = 8500 represents this situation, where x x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Using Natural Logarithms

The most frequently used base for logarithms is e . e . Base e e logarithms are important in calculus and some scientific applications; they are called natural logarithms . The base e e logarithm, log e ( x ) , log e ( x ) , has its own notation, ln ( x ) . ln ( x ) .

Most values of ln ( x ) ln ( x ) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, ln 1 = 0. ln 1 = 0. For other natural logarithms, we can use the ln ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e e using the inverse property of logarithms.

Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e . e . We write log e ( x ) log e ( x ) simply as ln ( x ) . ln ( x ) . The natural logarithm of a positive number x x satisfies the following definition.

We read ln ( x ) ln ( x ) as, “the logarithm with base e e of x x ” or “the natural logarithm of x . x . ”

The logarithm y y is the exponent to which e e must be raised to get x . x .

Since the functions y = e x y = e x and y = ln ( x ) y = ln ( x ) are inverse functions, ln ( e x ) = x ln ( e x ) = x for all x x and e = ln ( x ) x e = ln ( x ) x for x > 0. x > 0.

Given a natural logarithm with the form y = ln ( x ) , y = ln ( x ) , evaluate it using a calculator.

  • Press [LN] .

Evaluating a Natural Logarithm Using a Calculator

Evaluate y = ln ( 500 ) y = ln ( 500 ) to four decimal places using a calculator.

Rounding to four decimal places, ln ( 500 ) ≈ 6.2146 ln ( 500 ) ≈ 6.2146

Evaluate ln ( −500 ) . ln ( −500 ) .

Access this online resource for additional instruction and practice with logarithms.

  • Introduction to Logarithms

6.3 Section Exercises

What is a base b b logarithm? Discuss the meaning by interpreting each part of the equivalent equations b y = x b y = x and log b x = y log b x = y for b > 0 , b ≠ 1. b > 0 , b ≠ 1.

How is the logarithmic function f ( x ) = log b x f ( x ) = log b x related to the exponential function g ( x ) = b x ? g ( x ) = b x ? What is the result of composing these two functions?

How can the logarithmic equation log b x = y log b x = y be solved for x x using the properties of exponents?

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base b , b , and how does the notation differ?

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base b , b , and how does the notation differ?

For the following exercises, rewrite each equation in exponential form.

log 4 ( q ) = m log 4 ( q ) = m

log a ( b ) = c log a ( b ) = c

log 16 ( y ) = x log 16 ( y ) = x

log x ( 64 ) = y log x ( 64 ) = y

log y ( x ) = −11 log y ( x ) = −11

log 15 ( a ) = b log 15 ( a ) = b

log y ( 137 ) = x log y ( 137 ) = x

log 13 ( 142 ) = a log 13 ( 142 ) = a

log ( v ) = t log ( v ) = t

ln ( w ) = n ln ( w ) = n

For the following exercises, rewrite each equation in logarithmic form.

4 x = y 4 x = y

c d = k c d = k

m − 7 = n m − 7 = n

19 x = y 19 x = y

x − 10 13 = y x − 10 13 = y

n 4 = 103 n 4 = 103

( 7 5 ) m = n ( 7 5 ) m = n

y x = 39 100 y x = 39 100

10 a = b 10 a = b

e k = h e k = h

For the following exercises, solve for x x by converting the logarithmic equation to exponential form.

log 3 ( x ) = 2 log 3 ( x ) = 2

log 2 ( x ) = − 3 log 2 ( x ) = − 3

log 5 ( x ) = 2 log 5 ( x ) = 2

log 3 ( x ) = 3 log 3 ( x ) = 3

log 2 ( x ) = 6 log 2 ( x ) = 6

log 9 ( x ) = 1 2 log 9 ( x ) = 1 2

log 18 ( x ) = 2 log 18 ( x ) = 2

log 6 ( x ) = − 3 log 6 ( x ) = − 3

log ( x ) = 3 log ( x ) = 3

ln ( x ) = 2 ln ( x ) = 2

For the following exercises, use the definition of common and natural logarithms to simplify.

log ( 100 8 ) log ( 100 8 )

10 log ( 32 ) 10 log ( 32 )

2 log ( .0001 ) 2 log ( .0001 )

e ln ( 1.06 ) e ln ( 1.06 )

ln ( e − 5.03 ) ln ( e − 5.03 )

e ln ( 10.125 ) + 4 e ln ( 10.125 ) + 4

For the following exercises, evaluate the base b b logarithmic expression without using a calculator.

log 3 ( 1 27 ) log 3 ( 1 27 )

log 6 ( 6 ) log 6 ( 6 )

log 2 ( 1 8 ) + 4 log 2 ( 1 8 ) + 4

6 log 8 ( 4 ) 6 log 8 ( 4 )

For the following exercises, evaluate the common logarithmic expression without using a calculator.

log ( 10 , 000 ) log ( 10 , 000 )

log ( 0.001 ) log ( 0.001 )

log ( 1 ) + 7 log ( 1 ) + 7

2 log ( 100 − 3 ) 2 log ( 100 − 3 )

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

ln ( e 1 3 ) ln ( e 1 3 )

ln ( 1 ) ln ( 1 )

ln ( e − 0.225 ) − 3 ln ( e − 0.225 ) − 3

25 ln ( e 2 5 ) 25 ln ( e 2 5 )

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

log ( 0.04 ) log ( 0.04 )

ln ( 15 ) ln ( 15 )

ln ( 4 5 ) ln ( 4 5 )

log ( 2 ) log ( 2 )

ln ( 2 ) ln ( 2 )

Is x = 0 x = 0 in the domain of the function f ( x ) = log ( x ) ? f ( x ) = log ( x ) ? If so, what is the value of the function when x = 0 ? x = 0 ? Verify the result.

Is f ( x ) = 0 f ( x ) = 0 in the range of the function f ( x ) = log ( x ) ? f ( x ) = log ( x ) ? If so, for what value of x ? x ? Verify the result.

Is there a number x x such that ln x = 2 ? ln x = 2 ? If so, what is that number? Verify the result.

Is the following true: log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? Verify the result.

Is the following true: ln ( e 1.725 ) ln ( 1 ) = 1.725 ? ln ( e 1.725 ) ln ( 1 ) = 1.725 ? Verify the result.

Real-World Applications

The exposure index E I E I for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation E I = log 2 ( f 2 t ) , E I = log 2 ( f 2 t ) , where f f is the “f-stop” setting on the camera, and t t is the exposure time in seconds. Suppose the f-stop setting is 8 8 and the desired exposure time is 2 2 seconds. What will the resulting exposure index be?

Refer to the previous exercise. Suppose the light meter on a camera indicates an E I E I of − 2 , − 2 , and the desired exposure time is 16 seconds. What should the f-stop setting be?

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 − M 2 log I 1 I 2 = M 1 − M 2 where M M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. 8 How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

  • 4 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.
  • 5 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.
  • 6 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.
  • 7 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
  • 8 http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.

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Logarithmic Functions

The logarithmic function is an important medium of math calculations. Logarithms were discovered in the 16 th century by John Napier a Scottish mathematician, scientist, and astronomer. It has numerous applications in astronomical and scientific calculations involving huge numbers. Logarithmic functions are closely related to exponential functions and are considered as an inverse of the exponential function. The exponential function a x = N is transformed to a logarithmic function log a N = x.

The logarithm of any number N if interpreted as an exponential form, is the exponent to which the base of the logarithm should be raised, to obtain the number N. Here we shall aim at knowing more about logarithmic functions, types of logarithms, the graph of the logarithmic function, and the properties of logarithms.

What are Logarithmic Functions?

The basic logarithmic function is of the form f(x) = log a x (r) y = log a x, where a > 0. It is the inverse of the exponential function a y = x. Log functions include natural logarithm (ln) or common logarithm (log). Here are some examples of logarithmic functions:

  • f(x) = ln (x - 2)
  • g(x) = log 2 (x + 5) - 2
  • h(x) = 2 log x, etc.

Some of the non-integral exponent values can be calculated easily with the use of logarithmic functions. Finding the value of x in the exponential expressions 2 x = 8, 2 x = 16 is easy, but finding the value of x in 2 x = 10 is difficult. Here we can use log functions to transform 2 x = 10 into logarithmic form as log 2 10 = x and then find the value of x. The logarithm counts the number of occurrences of the base in repeated multiples. The formula for transforming an exponential function into a logarithmic function is as follows.

Logarithmic Function

The exponential function of the form a x = N can be transformed into a logarithmic function log a N = x. The logarithms are generally calculated with a base of 10, and the logarithmic value of any number can be found using a Napier logarithm table . The logarithms can be calculated for positive whole numbers, fractions, decimals, but cannot be calculated for negative values.

Domain and Range of Log Functions

Let us consider the basic (parent) common logarithmic function f(x) = log x (or y = log x). We know that log x is defined only when x > 0 (try finding log 0, log (-1), log (-2), etc using your calculator. You will come up with an error). So the domain is the set of all positive real numbers . Now, we will observe some of the y-values (outputs) of the function for different x-values (inputs).

  • When x = 1, y = log 1 = 0
  • When x = 2, y = log 2 = 0.3010
  • When x = 0.2, y = -0.6990
  • When x = 0.01, y = -2, etc

We can see that y can be either a positive or negative real number (or) it can be zero as well. Thus, y can take the value of any real number. Hence, the range of a logarithmic function is the set of all real numbers. Thus:

  • The domain of log function y = log x is x > 0 (or) (0, ∞).
  • The range of any log function is the set of all real numbers (R)

Example: Find the domain and range of the logarithmic function f(x) = 2 log (2x - 4) + 5.

For finding domain, set the argument of the function greater than 0 and solve for x.

2x - 4 > 0 2x > 4 x > 2

Thus, domain = (2, ∞).

As we have seen earlier, the range of any log function is R. So the range of f(x) is R.

Logarithmic Graph

We have already seen that the domain of the basic logarithmic function y = log a x is the set of positive real numbers and the range is the set of all real numbers. We know that the exponential and log functions are inverses of each other and hence their graphs are symmetric with respect to the line y = x. Also, note that y = 0 when x = 0 as y = log a 1 = 0 for any 'a'. Thus, all such functions have an x-intercept of (1, 0). A logarithmic function doesn't have a y-intercept as log a 0 is not defined. Summarizing all these, the graphs of exponential functions and logarithmic graph look like below.

Logarithmic Graph

Properties of Logarithmic Graph

  • a > 0 and a ≠ 1
  • The logarithmic graph increases when a > 1, and decreases when 0 < a < 1.
  • The domain is obtained by setting the argument of the function greater than 0.
  • The range is the set of all real numbers.

Graphing Logarithmic Functions

Before drawing a log function graph, just have an idea of whether you get an increasing curve or decreasing curve as the answer. If the base > 1, then the curve is increasing; and if 0 < base < 1, then the curve is decreasing. Here are the steps for graphing logarithmic functions :

  • Find the domain and range.
  • Find the vertical asymptote by setting the argument equal to 0. Note that a log function doesn't have any horizontal asymptote.
  • Substitute some value of x that makes the argument equal to 1 and use the property log a 1 = 0. This gives us the x-intercept.
  • Substitute some value of x that makes the argument equal to the base and use the property log a a = 1. This would give us a point on the graph.
  • Join the two points (from the last two steps) and extend the curve on both sides with respect to the vertical asymptote.

Example: Graph the logarithmic function f(x) = 2 log 3 (x + 1).

Here, the base is 3 > 1. So the curve would be increasing.

For domain: x + 1 > 0 ⇒ x > -1. So domain = (-1, ∞).

Vertical asymptote is x = -1.

  • At x = 0, y = 2 log 3 (0 + 1) = 2 log 3 1 = 2 (0) = 0
  • At x = 2, y = 2 log 3 (2 + 1)= 2 log 3 3 = 2 (1) = 2

If we want more clarity, we can form a table of values with some random values of x and substitute each of them in the given function to compute the y-values. This way, we get more points on the graph and it helps in getting the perfect shape of the graph.

Thus, (0, 0) and (2, 2) are two points on the curve. Thus, the log function graph looks as follows.

Graphing Logarithmic Functions

Properties of Logarithmic Functions

Logarithmic function properties are helpful to work across complex log functions. All the general arithmetic operations across numbers are transformed into a different set of operations within logarithms. The product of two numbers, when taken within the logarithmic functions is equal to the sum of the logarithmic values of the two functions. Similarly, the operations of division are transformed into the difference of the logarithms of the two numbers. Let us list the important properties of log functions in the below points.

  • log ab = log a + log b
  • loga/b = log a - log b
  • log b a = (log c a)/(log c b) ( change of base rule )
  • loga x = x loga
  • log a 1 = 0
  • log a a = 1

Derivative and Integral of Logarithmic Functions

The derivation of the logarithmic function gives the slope of the tangent to the curve representing the logarithmic function. The formula for the derivative of the common and natural logarithmic functions are as follows.

  • The derivative of ln x is 1/x. i.e., d/dx. ln x = 1/x.
  • The derivative of logₐ x is 1/(x ln a). i.e., d/dx (logₐ x) = 1/(x ln a).

The integral formulas of logarithmic functions are as follows:

  • The integral of ln x is ∫ ln x dx = x (ln x - 1) + C.
  • The integral of log x is ∫ log x dx = x (log x - 1) + C.

Related Topics:

  • Exponent Rules
  • Properties of Logarithms
  • Logs in calculations

Solved Examples on Logarithmic Functions

Example 1: Express 4 3 = 64 in logarithmic form.

Solution: The exponential form a x = N can be written in logarithmic function form as log a N = x .

Hence, 4 3 = 64 can be written in logarithmic form as log 4 64 = 3.

Answer: log 4 64 = 3

Example 2: Simplify log 2 (1/128).

Solution: We use the properties of logarithmic function to simplify the given logarithm.

log 2 (1/128) = log 2 1 - log 2 128

= 0 - log 2 2 7

= -log 2 2 7

= -7 log 2 2

Answer: Hence log 2 (1/128) = -7

Example 3: Find the domain, range, vertical and horizontal asymptotes of the logarithmic function f(x) = 3 log 2 (2x - 3) - 7.

For domain, 2x - 3 > 0 ⇒ x > 3/2. Hence domain = (3/2, ∞).

The range of any log function is (-∞, ∞).

For vertical asymptote (VA), 2x - 3 = 0 ⇒ x = 3/2.

A logarithmic graph never has a horizontal asymptote (HA).

Answer: Domain = (3/2, ∞); Range = (-∞, ∞); VA is x = 3/2; No HA.

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homework 5 graphing logarithmic functions

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Practice Questions on Logarithmic Functions

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FAQs on Logarithmic Functions

How to solve logarithmic functions.

The logarithmic function can be solved using the logarithmic formulas. The product of functions within logarithms is equal (log ab = log a + log b) to the sum of two logarithm functions. The division of two logarithm functions(loga/b = log a - log b) is changed to the difference of logarithm functions. The logarithm functions can also be solved by changing it to exponential form.

How to Graph Logarithmic Functions?

The graph of log function y = log x can be obtained by finding its domain, range, asymptotes, and some points on the curve. To find some points on the curve we can use the following properties:

What are Asymptotes of a Logarithmic Function?

Here are the asymptotes of a logarithmic function f(x) = a log (x - b) + c:

  • The vertical asymptote is x = b.
  • There is no horizontal asymptote .

How Are Exponential and Logarithmic Functions Related?

The exponential function of the form a x = N can be transformed into a logarithmic function log a N = x. Here the exponential functions 2 x = 10 is transformed into logarithmic form as log 2 10 = x, to find the value of x. The logarithm counts the numbers of occurrences of the base in repeated multiples.

What is the Difference Between Natural Logarithmic and Common Logarithmic Functions?

The logarithmic functions are broadly classified into two types, based on the base of the logarithms. We have natural logarithms and common logarithms. Natural logarithms are logarithms to the base 'e', and common logarithms are logarithms to the base of 10. Further logarithms can be calculated with reference to any base, but are often calculated for the base of either 'e' or '10'. The natural logarithms are written as log e x (or) ln x, and the common logarithms are written as log 10 x (or) log x. To obtain the value of x from natural logarithms, it is equal to the power to which e has to be raised to obtain x.

  • log e N = 2.303 ×log 10 N
  • log 10 N = 0.4343 × log e N

The value of e = 2.718281828459, but is often written in short as e = 2.718. Also, the above formulas help in the interconversion of natural logarithms and common logarithms.

How to Differentiate Logarithmic Functions?

The differentiation of a logarithmic function results in the inverse of the function. The differentiation of ln x is equal to 1/x. (d/dx .ln x = 1//x). Also, the antiderivative of 1/x gives back the ln function.

What Is the Range of Logarithmic Functions?

The range of a logarithmic function takes all values, which include the positive and negative real number values. Thus the range of the logarithmic function is from negative infinity to positive infinity.

What Is the Domain of Logarithmic Functions?

The logarithms can be calculated for positive whole numbers , fractions , decimals , but cannot be calculated for negative values. Hence the domain of the logarithmic function is the set of all positive real numbers.

What is the Formula for Logarithmic Functions?

The following formulas are helpful to work and solve the log functions.

  • log b a = (log a)/(log b)

What Are Logarithmic Functions Used For?

Logarithmic functions have numerous applications in physics, engineering, astronomy. The numeric measurements in astronomy include huge numbers with decimals and exponents. The huge scientific calculations can be easily simplified and calculated using log functions. The logarithmic functions help in transforming the product and division of numbers into sum and difference of numbers.

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3.5: Graphs and Properties of Logarithmic Functions

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  • Rupinder Sekhon and Roberta Bloom
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Learning Objectives

In this section, you will:

  • examine properties of logarithmic functions
  • examine graphs of logarithmic functions
  • examine the relationship between graphs of exponential and logarithmic functions

Recall that the exponential function \(f(x) = 2^x\) produces this table of values

Since the logarithmic function is an inverse of the exponential, \(g(x)=\log_{2}(x)\) produces the table of values

In this second table, notice that

  • As the input increases, the output increases.
  • As input increases, the output increases more slowly.
  • Since the exponential function only outputs positive values, the logarithm can only accept positive values as inputs, so the domain of the log function is \((0, \infty)\).
  • Since the exponential function can accept all real numbers as inputs, the logarithm can have any real number as output, so the range is all real numbers or \((-\infty, \infty)\).

5.5.png

Plotting the graph of \(g(x) = \log_{2}(x)\) from the points in the table , notice that as the input values for \(x\) approach zero, the output of the function grows very large in the negative direction, indicating a vertical asymptote at \(x = 0\).

In symbolic notation we write

as \(x \rightarrow 0^{+}\), \(f(x) \rightarrow-\infty\)

and as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

Source: The material in this section of the textbook originates from David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, “ Chapter 4: Exponential and Logarithmic Functions ,” licensed under a Creative Commons CC BY-SA 3.0 license. The material here is based on material contained in that textbook but has been modified by Roberta Bloom, as permitted under this license.

Graphically, in the function \(g(x) = \log_{b}(x)\), \(b > 1\), we observe the following properties:

5.5b.png

  • The graph has a horizontal intercept at (1, 0)
  • The line x = 0 (the y-axis) is a vertical asymptote; as \(x \rightarrow 0^{+}, y \rightarrow-\infty\)
  • The graph is increasing if \(b > 1\)
  • The domain of the function is \(x > 0\), or (0, \(\infty\))
  • The range of the function is all real numbers, or \((-\infty, \infty)\)

However if the base \(b\) is less than 1, 0 < \(b\) < 1, then the graph appears as below. This follows from the log property of reciprocal bases : \(\log _{1 / b} C=-\log _{b}(C)\)

5.5c.png

  • The line x = 0 (the y-axis) is a vertical asymptote; as \(x \rightarrow 0^{+}, y \rightarrow \infty\)
  • The graph is decreasing if 0 < \(b\) < 1
  • The domain of the function is \(x\) > 0, or (0, \(\infty\))

When graphing a logarithmic function, it can be helpful to remember that the graph will pass through the points (1, 0) and (\(b\), 1).

Finally, we compare the graphs of \(y = b^x\) and \(y = \log_{b}(x)\), shown below on the same axes.

Because the functions are inverse functions of each other, for every specific ordered pair (\(h\), \(k\)) on the graph of \(y = b^x\), we find the point (\(k\), \(h\)) with the coordinates reversed on the graph of \(y = \log_{b}(x)\).

In other words, if the point with \(x = h\) and \(y = k\) is on the graph of \(y = b^x\), then the point with \(x = k\) and \(y = h\) lies on the graph of \(y = \log_{b} (x)\)

The domain of \(y = b^x\) is the range of \(y = \log_{b} (x)\)

The range of \(y = b^x\) is the domain of \(y = \log_{b} (x)\)

For this reason, the graphs appear as reflections, or mirror images, of each other across the diagonal line \(y=x\). This is a property of graphs of inverse functions that students should recall from their study of inverse functions in their prerequisite algebra class.

5.5d.png

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Recall that the exponential function \(f(x)=2^{x}\) produces this table of values

Since the logarithmic function is an inverse of the exponential, \(g(x)=\log _{2} (x)\) produces the table of values

In this second table, notice that

1. As the input increases, the output increases. 2. As input increases, the output increases more slowly. 3. Since the exponential function only outputs positive values, the logarithm can only accept positive values as inputs, so the domain of the log function is \((0,\infty )\). 4. Since the exponential function can accept all real numbers as inputs, the logarithm can output any real number, so the range is all real numbers or \((-\infty ,\infty )\).

Sketching the graph, notice that as the input approaches zero from the right, the output of the function grows very large in the negative direction, indicating a vertical asymptote at \(x = 0\).

In symbolic notation we write as \(x \to 0^{+} ,f(x) \to -\infty\), and as\(x \to \infty ,f(x) \to \infty\)

A graph that approaches negative infinity as x approaches 0 from the right, and increases curving downwards passing through 1 comma 0, 2 comma 1, 4 comma 2 and 8 comma 3

Graphical Features of the Logarithm

  • Graphically, in the function \(g(x)=\log _{b} (x)\)
  • The graph has a horizontal intercept at (1, 0)
  • The graph has a vertical asymptote at \(x = 0\)
  • The graph is increasing and concave down
  • The domain of the function is \(x > 0\), or \((0,\infty )\)
  • The range of the function is all real numbers, or \((-\infty ,\infty )\)

When sketching a general logarithm with base \(b\), it can be helpful to remember that the graph will pass through the points (1, 0) and (\(b\), 1). To get a feeling for how the base affects the shape of the graph, examine the graphs below.

Three log graphs on the same axes, all that approach negative infinity as x approaches 0 from the right, and that increase curving downwards, all passing through 1 comma 0.  The first, labeled log of x, increases most slowly and passes through 10 comma 1.  The second, labeled ln of x, increases more rapidly, and passes through a point around 2.818 comma 1.  The third, labeled log base 2 of x, increases most rapidly, and passes through 2 comma 1.

Notice that the larger the base, the slower the graph grows. For example, the common log graph, while it grows without bound, it does so very slowly. For example, to reach an output of 8, the input must be 100,000,000.

Another important observation made was the domain of the logarithm. Like the reciprocal and square root functions, the logarithm has a restricted domain which must be considered when finding the domain of a composition involving a log.

Example \(\PageIndex{1}\)

Find the domain of the function \(f(x)=\log (5-2x)\).

The logarithm is only defined with the input is positive, so this function will only be defined when \(5 - 2x > 0\). Solving this inequality,

\[\begin{align*} -2x &>-5 \\[4pt] x &< \dfrac{5}{2} \end{align*} \nonumber\]

The domain of this function is \(x < \dfrac{5}{2}\), or in interval notation, \(\left(-\infty ,\dfrac{5}{2} \right)\)

Exercise \(\PageIndex{1}\)

Find the domain of the function \(f(x)=\log (x-5)+2\); before solving this as an inequality, consider how the function has been transformed.

Domain: {\(x\) | \(x > 5\)}

Transformations of the Logarithmic Function

Transformations can be applied to a logarithmic function using the basic transformation techniques, but as with exponential functions, several transformations result in interesting relationships.

First recall the change of base property tells us that:

\[\log _{b} x=\dfrac{\log _{c} x}{\log _{c} b} =\dfrac{1}{\log _{c} b} \log _{c} x \nonumber\]

From this, we can see that \(\log _{b} x\) is a vertical stretch or compression of the graph of the \(\log _{c} x\) graph. This tells us that a vertical stretch or compression is equivalent to a change of base. For this reason, we typically represent all graphs of logarithmic functions in terms of the common or natural log functions.

Next, consider the effect of a horizontal compression on the graph of a logarithmic function. Considering \(f(x)=\log (cx)\), we can use the sum property to see

\[ \begin{align*} f(x) &=\log (cx) \\[4pt] &=\log (c)+\log (x) \end{align*}\]

Since log(\(c\)) is a constant, the effect of a horizontal compression is the same as the effect of a vertical shift.

Example \(\PageIndex{2}\)

Sketch \(f(x)=\ln (x)\) and \(g(x)=\ln (x)+2\).

Two log graphs on the same axes, both which approach negative infinity as x approaches 0 from the right, and that increase curving downwards. The first labeled ln of x passes through 1 comma 0 and 2.818 comma 1.  The second labeled ln of x plus 2 passes through 1 comma 2 and 2.818 comma 3

Graphing these,

Note that this vertical shift could also be written as a horizontal compression, since \(g(x)=\ln (x)+2=\ln (x)+\ln (e^{2} )=\ln (e^{2} x).\)

While a horizontal stretch or compression can be written as a vertical shift, a horizontal reflection is unique and separate from vertical shifting.

Finally, we will consider the effect of a horizontal shift on the graph of a logarithm.

Example \(\PageIndex{3}\)

Sketch a graph of \(f(x)=\ln (x+2)\).

This is a horizontal shift to the left by 2 units. Notice that none of our logarithm rules allow us rewrite this in another form, so the effect of this transformation is unique. Shifting the graph,

A graph that approaches negative infinity as x approaches negative 2 from the right, and increases curving downwards passing through negative 1 comma 0 and 0.818 comma 1

Notice that due to the horizontal shift, the vertical asymptote shifted to \(x = -2\), and the domain shifted to \((-2,\infty )\).

Combining these transformations,

Example \(\PageIndex{4}\)

Sketch a graph of \(f(x) = 5\log(-x + 2)\).

A graph that decreases curving downwards, passing through negative 8 comma 5 and 1 comma 0 and approaches negative infinity as x approaches 2 from the left.

Factoring the inside as \(f (x) = 5\log(−(x − 2))\) reveals that this graph is that of the common logarithm, horizontally reflected, vertically stretched by a factor of 5, and shifted to the right by 2 units.

The vertical asymptote will be shifted to \(x = 2\) and the graph will have domain \((\infty, 2)\). A rough sketch can be created by using the vertical asymptote along with a couple points on the graph, such as

\[ \begin{align*} (f (1) &= 5\log(−1+ 2) = 5\log(1) = 0 \\[4pt] f (−8) &= 5\log(−(−8) + 2) = 5\log(10) = 5 \end{align*}\]

Exercise \(\PageIndex{2}\)

Sketch a graph of the function \(f(x)=-3\log (x-2)+1\).

A graph that approaches infinity as x approaches 2 from the right and decreases curving upwards, passing through 3 comma 1, 4 comma 0, and 12 comma negative 2

transormations of logs

Any transformed logarithmic function can be written in the form

\[f(x)=a\log (x-b)+k\text{ ,or }f(x)=a\log \left(-\left(x-b\right)\right)+k\] if horizontally reflected,

where \(x = b\) is the vertical asymptote.

Example \(\PageIndex{5}\)

Find an equation for the logarithmic function graphed.

A graph that approaches infinity as x approaches negative 2 from the right and decreases curving upwards, passing through negative 1 comma 1, 0 comma 0, and 2 comma negative 1

This graph has a vertical asymptote at \(x = –2\) and has been vertically reflected. We do not know yet the vertical shift (equivalent to horizontal stretch) or the vertical stretch (equivalent to a change of base). We know so far that the equation will have form

\[f(x)=-a\log (x+2)+k\nonumber\]

It appears the graph passes through the points (–1, 1) and (2, –1). Substituting in (–1, 1),

\[\begin{array}{l} {1=-a\log (-1+2)+k} \\ {1=-a\log (1)+k} \\ {1=k} \end{array}\nonumber\]

Next, substituting in (2, –1),

\[\begin{array}{l} {-1=-a\log (2+2)+1} \\ {-2=-a\log (4)} \\ {a=\dfrac{2}{\log (4)} } \end{array}\nonumber\]

This gives us the equation

\[f(x)=-\dfrac{2}{\log (4)} \log (x+2)+1. \nonumber\]

This could also be written as

\[f(x)=-2\log _{4} (x+2)+1. \nonumber\]

Exercise \(\PageIndex{3}\)

Write an equation for the function graphed here.

A graph that increases curving upwards, passing through negative 2 comma negative 2 and 2 comma 0, then approaches infinity as x approaches 3 from the left.

The graph is horizontally reflected and has a vertical asymptote at \(x = 3\), giving form \(f(x)=a\log \left(-\left(x-3\right)\right)+k\). Substituting in the point (2,0) gives \(0=a\log \left(-\left(2-3\right)\right)+k\), simplifying to \(k = 0\). Substituting in (-2,-2), \[-2=a\log \left(-\left(-2-3\right)\right)\nonumber\] so \[\dfrac{-2}{\log (5)} =a\nonumber\]The equation is \[f(x)=\dfrac{-2}{\log (5)} \log \left(-\left(x-3\right)\right)\nonumber\] or \[f(x)=-2\log _{5} \left(-\left(x-3\right)\right)\nonumber\]

Write the domain and range of the function graphed in Example 5, and describe its long run behavior.

Domain: {\(x\) | \(x\) > -2}, Range: all real numbers; As \(x\to -2^{+} ,f(x)\to \infty\)and as \(x\to \infty ,f(x)\to -\infty\).

Important Topics of this Section

  • Graph of the logarithmic function (domain and range)
  • Transformation of logarithmic functions
  • Creating graphs from equations
  • Creating equations from graphs

IMAGES

  1. Solved Date: Bell:_ Homework 5: Graphing Logarithmic

    homework 5 graphing logarithmic functions

  2. Logarithmic Functions and Their Graphs

    homework 5 graphing logarithmic functions

  3. 5.06 Graphing Logarithmic Functions

    homework 5 graphing logarithmic functions

  4. Graphing Logarithmic Functions Worksheet Form

    homework 5 graphing logarithmic functions

  5. Graph Logarithmic Functions Lesson by Algebra and Beyond

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  6. Graphing Logarithmic Functions Cheat Sheet by Scaffolded Math and Science

    homework 5 graphing logarithmic functions

VIDEO

  1. Graphing Logarithmic Functions Examples

  2. GRAPHS OF LOGARITHMIC FUNCTIONS

  3. Graphing Logarithmic Functions

  4. Graphing Logarithmic Functions

  5. College Algebra Homework

  6. Introduction to graphs of logs and exponential function

COMMENTS

  1. 5.5: Graphs of Logarithmic Functions

    Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function ln(x) has base e ≈ 2.718 .) Figure 5.5.4: The graphs of three logarithmic functions with different bases, all greater than 1. Given a logarithmic function with the form f(x) = logb(x), graph the function.

  2. Solved Date: Bell:_ Homework 5: Graphing Logarithmic

    Math Algebra Algebra questions and answers Date: Bell:_ Homework 5: Graphing Logarithmic Functions ** This is a 2-page document! ** Directions: Graph each function and identify its key characteristics. 1.

  3. 1.3: Graphs of Logarithmic Functions

    The logarithmic function is defined only when the input is positive, so this function is defined when 5- 2x > 0 . Solving this inequality, 5 − 2x > 0 The input must be positive − 2x > − 5 Subtract 5 x < 5 2 Divide by -2 and switch the inequality. The domain of f(x) = log(5 − 2x) is (- ∞, 5 2). Exercise 1.3.2.

  4. 4.4: Graphs of Logarithmic Functions

    Previously, the domain and vertical asymptote were determined by graphing a logarithmic function. It is also possible to determine the domain and vertical asymptote of any logarithmic function algebraically. Here we will take a look at the domain (the set of input values) for which the logarithmic function is defined, and its vertical asymptote.

  5. 6.4 Graphs of Logarithmic Functions

    Graph logarithmic functions. In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations?

  6. Graphs of logarithmic functions (video)

    About Transcript The graph of y=log base 2 of x looks like a curve that increases at an ever-decreasing rate as x gets larger. It becomes very negative as x approaches 0 from the right. The graph of y=-log base 2 of x is the same as the first graph, but flipped over the x-axis.

  7. 3.4.1: Graphs of Logarithmic Functions

    To graph a logarithmic function using a TI-83/84, enter the function into Y= and use the Change of Base Formula: loga x = logb x logb a log a x = log b x log b a. The keystrokes would be: Y = log(x) log(1 4) + 2 Y = log ( x) log ( 1 4) + 2, GRAPH. To see a table of values, press 2 nd → GRAPH. Example 3. Graph y=−logx using a graphing ...

  8. Exponential & logarithmic functions

    This topic covers: - Radicals & rational exponents - Graphs & end behavior of exponential functions - Manipulating exponential expressions using exponent properties - Exponential growth & decay - Modeling with exponential functions - Solving exponential equations - Logarithm properties - Solving logarithmic equations - Graphing logarithmic functions - Logarithmic scale

  9. Graphing logarithmic functions (example 1)

    For y=(x-3)^2, where does y=0? At x=3. I.e., in the new graph, the old vertex (which was at x=0) is now at x=3, hence the graph has shifted 3 to the right. Similar principle in this video's equations -- we now have to change x to '-3' to get the same result as when x was '0' before, so the new function has moved over 3 to the left.

  10. Unit 7 Lesson 5

    This project was created with Explain Everything™ Interactive Whiteboard for iPad.

  11. 4.5: Graphing Logarithmic Functions

    MATH 141: Precalculus I (2nd Edition) 4: Exponential and Logarithmic Functions

  12. Algebra

    Here is the definition of the logarithm function. If b is any number such that b > 0 and b ≠ 1 and x > 0 then, y = logbx is equivalent to by = x. We usually read this as "log base b of x ". In this definition y = logbx is called the logarithm form and by = x is called the exponential form. Note that the requirement that x > 0 is really a ...

  13. Log function graph: log(x)

    Free online graphing calculator - graph functions, conics, and inequalities interactively

  14. 6.3 Logarithmic Functions

    Also, since the logarithmic and exponential functions switch the x x and y y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore, the domain of the logarithm function with base b is (0, ∞). b is (0, ∞). the range of the logarithm function with base b is (− ∞, ∞). b is (− ...

  15. Graphing Logarithmic Functions ( Read )

    The general form of a logarithmic function is f(x) = alogb(x − h) + k and the vertical asymptote is x = h. The domain is x > h and the range is all real numbers. Lastly, if b > 1, the graph moves up to the right. If 0 < b < 1, the graph moves down to the right. Let's graph y = log3(x − 4) and state the domain and range.

  16. Solved Name: Date: Unit 7: Exponential & Logarithmic

    1. f (x) = logg Domain: Range: End Behavior: As x As - - f (x) -- x-intercept: Asymptote: 2. f (x) = log1 Domain: Range: End Behavior: AS - (x) As 29 This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. See Answer

  17. Logarithmic Functions

    The basic logarithmic function is of the form f (x) = log a x (r) y = log a x, where a > 0. It is the inverse of the exponential function a y = x. Log functions include natural logarithm (ln) or common logarithm (log). Here are some examples of logarithmic functions: f (x) = ln (x - 2) g (x) = log 2 (x + 5) - 2 h (x) = 2 log x, etc.

  18. 5.06 Graphing with Logarithmic Functions

    The pool company developed new chemicals that transform the pH scale. Using the pH function f(x) = −log 10 x as the parent function, explain which transformation results in a y-intercept and why. You may graph by hand or using technology. Be sure to label each transformation on the graph. f(x) + 1 = -log 10 x + 1 f(x + 1) = -log 10 (x+1)

  19. 4.5: Graphs of Logarithmic Functions

    The logarithmic function is defined only when the input is positive, so this function is defined when 5- 2x > 0 . Solving this inequality, 5 − 2x > 0 The input must be positive − 2x > − 5 Subtract 5 x < 5 2 Divide by -2 and switch the inequality. The domain of f(x) = log(5 − 2x) is (- ∞, 5 2). Exercise 4.5.2.

  20. 3.5: Graphs and Properties of Logarithmic Functions

    The graph is increasing if b > 1. The domain of the function is x > 0, or (0, ∞) The range of the function is all real numbers, or ( − ∞, ∞) However if the base b is less than 1, 0 < b < 1, then the graph appears as below. This follows from the log property of reciprocal bases : log 1 / b C = − log b ( C)

  21. 5.5.1: Graphs and Properties of Logarithmic Functions (Exercises)

    Last updated Jul 17, 2022 5.5: Graphs and Properties of Logarithmic Functions 5.6: Application Problems with Exponential and Logarithmic Functions Rupinder Sekhon and Roberta Bloom De Anza College Table of contents SECTION 5.5 PROBLEM SET: GRAPHS AND PROPERTIES OF LOGARITHMIC FUNCTIONS

  22. Desmos

    Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

  23. 4.5: Graphs of Logarithmic Functions

    Since the logarithmic function is an inverse of the exponential, g(x) = log2(x) g ( x) = log 2 ( x) produces the table of values