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.

$2^{-x}=1.5$

Short answer

$x=-{\log }_{2}1.5=-\frac{\ln 1.5}{\ln 2}\approx -0.585$

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

$2^{-x}=1.5$

Step 2. Explanation

${\begin{array}{l}{\text{Wecansolveusingformulalog}}_a{a}^x=x\\ \Rightarrow 2^{-x}=1.5\\ \Rightarrow {\log }_2{2}^{-x}={\log }_{2}1.5\\ \Rightarrow -x={\log }_{2}1.5\\ \Rightarrow x=-{\log }_{2}1.5\\ {\text{Nowusingformulalog}}_{a}M=\frac{{\log }_{a}M}{{\log }_{b}a}\\ \text{Forbwewilltakeb=e}\\ \text{So,}\\ x=-{\log }_{2}1.5=-\frac{\ln 1.5}{\ln 2}\approx -0.585\end{array}}_{}$

Step 3. Verifying the result.

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

Hence,

The solution of the equation is$x=-0.585$