Question

Solve for $x$:

${\log }_7(x+2)+{\log }_7(x-3)={\log }_{7}24$

Short answer

The value of$x$is $\frac{74}{23}$.

Step-by-step solution

Step 1. Given Information 

We are given an equation ${\log }_7(x+2)+{\log }_7(x-3)={\log }_{7}24$.

Step 2. Solving the equation

Using quotients property of logarithm, we get

${\log }_7\frac{x+2}{x-3}={\log }_{7}24$

By comparing, we get

$\frac{x+2}{x-3}=24$

Step 3. Finding the value of x

Simplifying, we get

$$\begin{gathered} \frac{x+2}{x-3}=24 \\ 24(x-3)=x+2 \\ 24x-72=x+2 \\ 24x-x=72+2 \\ 23x=74 \\ x=\frac{74}{23} \end{gathered}$$