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

$0.3\left(4^{0.2x}\right)=0.2$

Short answer

$x=5.{\log }_4\frac{2}{3}=5\cdot \frac{\ln \frac{2}{3}}{\ln 4}\approx -1.462$

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.

$0.3\left(4^{0.2x}\right)=0.2$

Step 2. Explanation

$\begin{array}{l}{\text{Wecansolveusingformulalog}}_a{a}^x=x\\ \text{Herea=4}\\ \text{Soweget}\\ \Rightarrow 4^{0.2x}=\frac{2}{3}\\ \Rightarrow {\log }_4{4}^{0.2x}={\log }_4\frac{2}{3}\\ \Rightarrow 0.2x={\log }_4\frac{2}{3}\\ \Rightarrow \frac{2}{10}x={\log }_4\frac{2}{3}\\ \Rightarrow \frac{1}{5}x={\log }_4\frac{2}{3}\\ \Rightarrow x=5.{\log }_4\frac{2}{3}\\ {\text{Nowusingformulalog}}_{a}M=\frac{{\log }_{a}M}{{\log }_{b}a}\\ \text{Forbwewilltakeb=e}\\ \text{So,}\\ x=5.{\log }_4\frac{2}{3}=5\cdot \frac{\ln \frac{2}{3}}{\ln 4}\approx -1.462\end{array}$

Step 3. Verify the solution with the graph.

The graph of the given function is:

Here, we can see that the graph is a straight line passes through the point $x=-1.462$.

Thus it is verified that the solution is correct.