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.

$5^{-x}=25$

Short answer

The solution of the given equation is x=-2

Step-by-step solution

Step 1. Given function

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.

$5^{-x}=25$

Step 2. Explanation

$\begin{array}{l}\Rightarrow 5^{-x}=25\\ \Rightarrow \ln 5^{-x}=\ln 5^2\\ \Rightarrow \left(-x\right)\cancel{\ln 5}=2\cancel{\ln 5}\\ \Rightarrow -x=2\\ \Rightarrow x=-2\end{array}$

Step 3. Verifying the result 

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

From the graph, we can see the line intersect at the x-axis at x= -2

Hence verified