Question

In the following exercises, solve each logarithmic equation.

${\log }_4\left(5x+6\right)=3$

Short answer

The solution is $x=11.6$.

Step-by-step solution

Step 1. Rewrite in exponential form and simplify.

Consider the logarithmic equation ${\log }_4\left(5x+6\right)=3$. Its equivalent exponential form is given as

$4^3=5x+6$

On simplifying we get,

$64=5x+6$

Step 2. Solve for x

Subtract $6$from both sides.

$\begin{array}{rcl}64-6& =& 5x+6-6\\ 58& =& 5x\end{array}$

Divide both sides by $5$.

$\begin{array}{rcl}\frac{58}{5}& =& \frac{5x}{5}\\ 11.6& =& x\end{array}$

Step 3. Check the solution

Substitute $11.6$for $x$in the original equation, we get

$\begin{array}{rcl}{\log }_4\left(5x+6\right)& =& 3\\ {\log }_4\left(5\times 11.6+6\right)& =& 3\\ {\log }_4\left(58+6\right)& =& 3\\ {\log }_4\left(64\right)& =& 3\\ 4^3& =& 64\\ 64& =& 64\end{array}$

So we get a true statement, thus the found solution is correct.