<meta http-equiv="refresh" content="1; url=/nojavascript/">
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra II with Trigonometry Concepts Go to the latest version.

8.6: Inverse Properties of Logarithmic Functions

Difficulty Level: At Grade Created by: CK-12
%
Progress
Practice Inverse Properties of Logarithms
Practice
Progress
%
Practice Now

If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation y' = y is the general function y = Ce^x . What is the inverse of this function?

Guidance

By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed (see the Inverse of a Function concept), they equal x . Therefore, if f(x)=b^x and g(x)=\log_b x , then:

f \circ g=b^{\log_b x}=x and g \circ f =\log_b b^x=x

These are called the Inverse Properties of Logarithms.

Example A

Find:

a) 10^{\log 56}

b) e^{\ln6} \cdot e^{\ln2}

Solution: For each of these examples, we will use the Inverse Properties.

a) Using the first property, we see that the bases cancel each other out. 10^{\log 56}=56

b) Here, e and the natural log cancel out and we are left with 6 \cdot 2=12 .

Example B

Find \log_4 16^x

Solution: We will use the second property here. Also, rewrite 16 as 4^2 .

\log_4 16^x=\log_4 (4^2)^x=\log_4 4^{2x}=2x

Example C

Find the inverse of f(x)=2e^{x-1} .

Solution: See the Finding the Inverse concept for the steps on how to find the inverse.

Change f(x) to y . Then, switch x and y .

& y=2e^{x-1} \\& x=2e^{y-1}

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.

& \frac{x}{2}=e^{y-1} \\& \ln \left(\frac{x}{2}\right)= \ln e^{y-1}

Recall the Inverse Properties from earlier in this concept. \log_b b^x=x ; applying this to the right side of our equation, we have \ln e^{y-1}=y-1 . Solve for y .

& \ln \left(\frac{x}{2}\right)=y-1 \\& \ln \left(\frac{x}{2}\right)+1=y

Therefore, \ln \left(\frac{x}{2}\right)+1 is the inverse of 2e^{y-1} .

Intro Problem Revisit Switch x and y in the function y = Ce^x and then solve for y .

x = Ce^y\\\frac{x}{C} = e^y\\ln \frac{x}{C} = ln (e^y)\\ln \frac{x}{C} = y

Therefore, the inverse of y = Ce^x is y = ln \frac{x}{C} .

Guided Practice

1. Simplify 5^{\log_5 6x} .

2. Simplify \log_9 81^{x+2} .

3. Find the inverse of f(x)=4^{x+2}-5 .

Answers

1. Using the first inverse property, the log and the base cancel out, leaving 6x as the answer.

5^{\log_5 6x}=6x

2. Using the second inverse property and changing 81 into 9^2 we have:

\log_9 81^{x+2} &= \log_9 9^{2(x+2)} \\&= 2(x+2) \\&= 2x+4

3. Follow the steps from Example C to find the inverse.

f(x) &= 4^{x+2}-5 \\y &= 4^{x+2}-5 \\x &= 4^{y+2}-5 \\x+5 &= 4^{y+2} \\\log_4 (x+5) &= y+2 \\\log_4 (x+5)-2 &= y

Vocabulary

Inverse Properties of Logarithms
\log_b b^x=x and b^{\log_b x}=x, b \ne 1

Practice

Use the Inverse Properties of Logarithms to simplify the following expressions.

  1. \log_3 27^x
  2. \log_5 \left(\frac{1}{5}\right)^x
  3. \log_2 \left(\frac{1}{32}\right)^x
  4. 10^{\log(x+3)}
  5. \log_6 36^{(x-1)}
  6. 9^{\log_9(3x)}
  7. e^{\ln(x-7)}
  8. \log \left(\frac{1}{100}\right)^{3x}
  9. \ln e^{(5x-3)}

Find the inverse of each of the following exponential functions.

  1. y=3e^{x+2}
  2. f(x)=\frac{1}{5}e^\frac{x}{7}
  3. y=2+e^{2x-3}
  4. f(x)=7^{\frac{3}{x}+1-5}
  5. y=2(6)^\frac{x-5}{2}
  6. f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}

Vocabulary

Inverse Properties of Logarithms

Inverse Properties of Logarithms

The inverse properties of logarithms are \log_b b^x=x and b^{\log_b x}=x, b \ne 1.

Image Attributions

Description

Difficulty Level:

At Grade

Grades:

Date Created:

Mar 12, 2013

Last Modified:

Apr 23, 2015
Files can only be attached to the latest version of Modality

Reviews

Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALY.334.3.L.1

Original text