Q. 74 - Precalculus Enhanced with Graphing Utilities

Question

solve each equation. Verify your results using a graphing utility.

$5^{x^2+8}={125}^{2x}$

Step-by-Step Solution

Answer

The value of x for which the above equation is valid are :

$2,4$

Step 1. Given equation

$5^{x^2+8}={125}^{2x}$

Step 2. Solving exponentially

$$\begin{gathered} 5^{x^2+8}={125}^{2x} \\ {5}^{x^2+8}=5^{3\left(2x\right)} \\ {5}^{x^2+8}=5^{6x} \end{gathered}$$

By exponential property 

$$\begin{gathered} a^v=a^u \\ \Rightarrow v=u \end{gathered}$$

Using this :

$$\begin{gathered} x^2+8=6x \\ \Rightarrow x^2-6x+8=0 \\ \Rightarrow x^2-4x-2x+8=0 \\ \Rightarrow x(x-4)-2(x-4)=0 \\ \Rightarrow (x-2)(x-4)=0 \\ \Rightarrow x=2 or x=4 \end{gathered}$$