Q. 65 - Precalculus Enhanced with Graphing Utilities

Question

Solve the equation ${\left(\frac{1}{5}\right)}^x=\frac{1}{25}$

and verify your results using a graphing utility.

Step-by-Step Solution

Answer

Required solution set is {2}. 

Step 1. Given information

${\left(\frac{1}{5}\right)}^x=\frac{1}{25}$

Step 2. Calculations

$$\begin{gathered} {\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)}^x=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right. \\ {5}^{-x}=5^{-2} if a^u=a^v then u=v \\ x=2 \end{gathered}$$

Step 3. Graphing solution

Graph$y={\frac{1}{5}}^x$  and y =1/25 intersect at (2,0.04)

Hence the required solution set is {2}.