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8.6: Inverse Properties of Logarithmic Functions

Difficulty Level: At Grade Created by: CK-12
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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=Cex. 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)=bx and g(x)=logbx, then:

fg=blogbx=x and gf=logbbx=x

These are called the Inverse Properties of Logarithms.

Example A

Find:

a) 10log56

b) eln6eln2

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. 10log56=56

b) Here, e and the natural log cancel out and we are left with 62=12.

Example B

Find log416x

Solution: We will use the second property here. Also, rewrite 16 as 42.

log416x=log4(42)x=log442x=2x

Example C

Find the inverse of f(x)=2ex1.

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=2ex1x=2ey1

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

x2=ey1ln(x2)=lney1

Recall the Inverse Properties from earlier in this concept. logbbx=x; applying this to the right side of our equation, we have lney1=y1. Solve for y.

ln(x2)=y1ln(x2)+1=y

Therefore, ln(x2)+1 is the inverse of 2ey1.

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

x=CeyxC=eylnxC=ln(ey)lnxC=y

Therefore, the inverse of y=Cex is y=lnxC.

Guided Practice

1. Simplify 5log56x.

2. Simplify log981x+2.

3. Find the inverse of f(x)=4x+25.

Answers

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

5log56x=6x

2. Using the second inverse property and changing 81 into 92 we have:

log981x+2=log992(x+2)=2(x+2)=2x+4

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

f(x)yxx+5log4(x+5)log4(x+5)2=4x+25=4x+25=4y+25=4y+2=y+2=y

Vocabulary

Inverse Properties of Logarithms
logbbx=x and blogbx=x,b1

Practice

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

  1. log327x
  2. log5(15)x
  3. log2(132)x
  4. 10log(x+3)
  5. log636(x1)
  6. 9log9(3x)
  7. eln(x7)
  8. log(1100)3x
  9. \begin{align*}\ln e^{(5x-3)}\end{align*}

Find the inverse of each of the following exponential functions.

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

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Vocabulary

Inverse Properties of Logarithms

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

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Date Created:
Mar 12, 2013
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Sep 07, 2016
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MAT.ALY.334.3.L.1