Question

Solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility:$5^x=3^{x+2}$

Short answer

The solution set of$x=\left\{\frac{2\ln 3}{\ln 5-\ln 3}\right\}\approx \left\{4.301\right\}$

Step-by-step solution

Step 1. Given information 

An equation,$5^x=3^{x+2}$

Step 2. Required to find 

Solve the given equation to find x.

Step 3. Solving the equation

Given, $5^x=3^{x+2}$

Taking log on both sides,

$\ln 5^x=\ln 3^{x+2}$

$\Rightarrow x\ln 5=(x+2)\ln 3$

$\Rightarrow x\ln 5-x\ln 3=2\ln 3$

$\Rightarrow x(\ln 5-\ln 3)=2\ln 3$

$\Rightarrow x=\frac{2\ln 3}{\ln 5-\ln 3}$

$\Rightarrow x=4.301$