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=Cex. What is the inverse of this function?

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 $$\ 2 e^{y-1}$$.

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}

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.

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

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