Question

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

$8^{-x}=1.2$

Short answer

$x=-{\log }_{8}1.2=\frac{-\ln 1.2}{\ln 8}\approx -0.088$

Step-by-step solution

Step 1. Given information

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

$8^{-x}=1.2$

Step 2. Explanastion

$\begin{array}{l}\Rightarrow 8^{-x}=1.2\\ \Rightarrow {\log }_8{8}^{-x}={\log }_{8}1.2\\ \Rightarrow -x={\log }_8\text{1.2}\\ \Rightarrow x=-{\log }_{8}1.2\\ \Rightarrow x=-{\log }_{8}1.2=\frac{-\ln 1.2}{\ln 8}\approx -0.088\end{array}$

Step 3. Verifying the result  

Plot the graph of the equation using the graphing utility as shown below

Hence,

The solution is$x=-0.088$