5. $(05.02)$
Explain how to solve $5^{x-2}=8$ using the change of base formula $\log _{b} y=\frac{\log y}{\log b}$. Include the solution for $x$ in your answer. Round your answer to the nearest thousandth. (10 points)
Answer
\(\boxed{x \approx 3.292}\)
Step 1 ：Given the equation \(5^{x-2}=8\)
Step 2 ：Take the logarithm of both sides of the equation to get \(\log(5^{x-2}) = \log(8)\)
Step 3 ：Use the properties of logarithms to simplify the left side of the equation to get \((x-2)\log(5) = \log(8)\)
Step 4 ：Divide both sides of the equation by \(\log(5)\) to isolate \(x-2\) on one side of the equation to get \(x-2 = \frac{\log(8)}{\log(5)}\)
Step 5 ：Add 2 to both sides of the equation to solve for \(x\) to get \(x = \frac{\log(8)}{\log(5)} + 2\)
Step 6 ：Calculate the value of \(x\) to get \(x \approx 3.292\)
Step 7 ：\(\boxed{x \approx 3.292}\)
