3.3.3: Inverse Properties of Logarithms

Inverse Properties of Logarithmic Functions

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?

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Inverse Properties of Logarithms

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, they equal \(\ x\). Therefore, if \(\ f(x)=b^{x} \text { and } g(x)=\log _{b} x\), then:

\(\ f \circ g=b^{\log _{b} x}=x \text { and } g \circ f=\log _{b} b^{x}=x\)

These are called the Inverse Properties of Logarithms.

Let's solve the following problems. We will use the Inverse Properties of Logarithms.

1. Find \(\ 10^{\log 56}\).

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

   \(\ e^{\ln 6} \cdot e^{\ln 2}\)

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

2. Find \(\ \log _{4} 16^{x}\) We will use the second property here. Also, rewrite 16 as 4².

   \(\ \log _{4} 16^{x}=\log _{4}\left(4^{2}\right)^{x}=\log _{4} 4^{2 x}=2 x\)

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

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

   \(\ \begin{array}{l}
   y=2 e^{x-1} \\
   x=2 e^{y-1}
   \end{array}\)

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

   \(\ \begin{array}{l}
   \frac{x}{2}=e^{y-1} \\
   \ln \left(\frac{x}{2}\right)=\ln e^{y-1}
   \end{array}\)

   Recall the Inverse Properties of Logarithms 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 \(\ y\).

   \(\ \begin{array}{l}
   \ln \left(\frac{x}{2}\right)=y-1 \\
   \ln \left(\frac{x}{2}\right)+1=y
   \end{array}\)

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

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Examples

Example 1

Earlier, you were asked to find the inverse of \(\ y=C e^{x}\).

Solution

Switch x and y in the function \(\ y=C e^{x}\) and then solve for y.

\(\ \begin{array}{r}
x=C e^{y} \\
\frac{x}{C}=e^{y} \\
\ln \frac{x}{C}=\ln \left(e^{y}\right) \\
\ln \frac{x}{C}=y
\end{array}\)

Therefore, the inverse of \(\ y=C e^{x} \text { is } y=\ln \frac{x}{C}\).

Example 2

Simplify \(\ 5^{\log _{5} 6 x}\).

Solution

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

\(\ 5^{\log _{5} 6 x}=6 x\)

Example 3

Simplify \(\ \log _{9} 81^{x+2}\).

Solution

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

\(\ \begin{aligned}
\log _{9} 81^{x+2} &=\log _{9} 9^{2(x+2)} \\
&=2(x+2) \\
&=2 x+4
\end{aligned}\)

Example 4

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

Solution

\(\ \begin{aligned}
f(x) &=4^{x+2}-5 \\
y &=4^{x+2}-5 \\
x &=4^{y+2}-5 \\
x+5 &=4^{y+2} \\
\log _{4}(x+5) &=y+2 \\
\log _{4}(x+5)-2 &=y
\end{aligned}\)

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Review

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. \(\ 9^{\log _{9}(3 x)}\)
7. \(\ e^{\ln (x-7)}\)
8. \(\ \log \left(\frac{1}{100}\right)^{3 x}\)
9. \(\ \ln e^{(5 x-3)}\)

Find the inverse of each of the following exponential functions.

10. \(\ y=3 e^{x+2}\)
11. \(\ f(x)=\frac{1}{5} e^{\frac{x}{7}}\)
12. \(\ y=2+e^{2 x-3}\)
13. \(\ f(x)=7^{\frac{3}{x}+1-5}\)
14. \(\ y=2(6)^{\frac{x-5}{2}}\)
15. \(\ f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}\)

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Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.6.

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Vocabulary

Term	Definition
Inverse Properties of Logarithms	The inverse properties of logarithms are \(\ \log _{b} b^{x}=x \text { and } b^{\log _{b} x}=x, b \neq 1\).