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.

$2^{x-5}=8$

Short answer

The solution of the given equation is x=8

Step-by-step solution

Step 1. Given Function

The given equation is $2^{x-5}=8$.

We have to express irrational solutions in exact form and as a decimal rounded to three decimal places.

Also, We have to verify the result using a graphing utility.

Step 2. Explanation

$\begin{array}{l}\Rightarrow 2^{x-5}=8\\ \Rightarrow \ln 2^{x-5}=\ln 2^3\\ \Rightarrow \left(x-5\right)\cancel{\ln 2}=3\cancel{\ln 2}\\ \Rightarrow x-5=3\\ \Rightarrow x-5+5=3+5\left[\text{Add}5\text{boththesides}\right]\\ \Rightarrow x=8\end{array}$

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=8

Hence verified