Question

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

${\left(\frac{1}{5}\right)}^x=9$

Short answer

The exact answer is $-1.365212$and approximated answer is $-1.365$.

Step-by-step solution

Step 1. Given Information 

We are given an expression ${\left(\frac{1}{5}\right)}^x=9$.

Use properties of logarithm to solve the given equation.

Step 2. Solving the expression

Taking logarithm on both sides, we get

$\ln {\left(\frac{1}{5}\right)}^x=\ln \left(9\right)$

Using power property of logarithm, we get

$$\begin{gathered} x\ln \left(\frac{1}{5}\right)=\ln \left(9\right) \\ x=\frac{\ln \left(9\right)}{\ln \left(\frac{1}{5}\right)} \\ x=-1.365212 \\ x=-1.365 \end{gathered}$$