Question

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

${\log }_2\left(5x\right)=4$

Short answer

The solution of the logarithmic equation${\log }_2\left(5x\right)=4$is localid="1646320214453" $\left\{\frac{16}{5}\right\}or3.2$

Step-by-step solution

Step 1. Given information

$\text{Thegivenequationis}{\log }_2\left(5x\right)=4.$

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

Step 2. Simplify.

$\begin{array}{l}\text{Now,}\\ \text{Wecanwrite,}\\ \Rightarrow {\text{log}}_{a}b=c\text{as}b=a^c.\\ \text{Thus,}\\ \Rightarrow {\log }_2\left(5x\right)=4\\ \Rightarrow 5x=2^4\\ \Rightarrow 5x=16\\ \Rightarrow x=\frac{16}{5}=3.2\left[\text{Dividebothsidesby}5.\right]\end{array}$

Step 3. Verifying the result

Plot the graph of the equation ${\log }_2\left(5x\right)=4$using the graphing utility.

From the above graph. we can observe that the graph of the given equation intersects the x-axis at $x=\frac{16}{5}$.

Therefore, the solution is verified.