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.

${\log }_5\left(2x+3\right)={\log }_{5}3$

Short answer

The solution of the given logarithmic equation is$\left\{0\right\}.$

Step-by-step solution

Step 1. Given information

$\text{Thegivenequationis}{\log }_5\left(2x+3\right)={\log }_{5}3.$

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 the graphing utility.

Step 2. Simplify.

$\begin{array}{l}\text{Now,}\\ \text{Weknow,}\\ \Rightarrow {\text{log}}_{a}b=\frac{\log b}{\log a}.\\ \text{Thus,}\\ \Rightarrow {\log }_5\left(2x+3\right)={\log }_{5}3\\ \Rightarrow \frac{\log \left(2x+3\right)}{\cancel{\log 5}}=\frac{\log 3}{\cancel{\log 5}}\\ \Rightarrow \log \left(2x+3\right)=\log 3\\ \Rightarrow 2x+3=3\\ \Rightarrow 2x=3-3\\ \Rightarrow 2x=0\\ \Rightarrow x=0\end{array}$

Step 3. Verifying the result

Plot the graph of the equation ${\log }_5\left(2x+3\right)={\log }_{5}3$using the graphing utility.

From the above graph. we can observe that the graph of the given equation intersects the x-axis at x=0.

Therefore, the solution is verified.