Question

In the following exercises, solve each logarithmic equation.

${\log }_2\left(5x-7\right)=3$

Short answer

The solution is $x=3$.

Step-by-step solution

Step 1. Convert into exponential form

Consider the logarithmic equation ${\log }_2\left(5x-7\right)=3$.

Its equivalent exponential form is given as $2^3=5x-7$.

And it is simplified to

$8=5x-7$

Step 2. Solve the equation for x

To solve the equation add $7$both sides.

$\begin{array}{rcl}8+7& =& 5x-7+7\\ 15& =& 5x\end{array}$

Now divide both sides by $5$.

$\begin{array}{rcl}\frac{15}{5}& =& \frac{5x}{5}\\ 3& =& x\end{array}$

Step 3. Check the solution

Substitute $3$for $x$in the original equation.

$\begin{array}{rcl}{\log }_2\left(5x-7\right)& =& 3\\ {\log }_2\left(5\times 3-7\right)& =& 3\\ {\log }_2\left(15-7\right)& =& 3\\ {\log }_2\left(8\right)& =& 3\\ 2^3& =& 8\\ 8& =& 8\end{array}$

As the statement is true, so the found solution is correct.