Question

In the following exercises, solve for x.

${\log }_x\left(125\right)=3$

Short answer

The solution is $x=5$.

Step-by-step solution

Step 1. Convert into exponential form. 

Consider the logarithmic equation ${\log }_x\left(125\right)=3$.

Its equivalent exponential form is given as $x^3=125$.

Step 2. Compare the bases

The number $125$can be written as $5^3$. So the exponential equation can be written as

$$\begin{gathered} x^3=125 \\ {x}^3=5^3 \end{gathered}$$

Now as the exponents are the same, so on comparing the bases we get

$x=5$

Step 3. Check the solution

Substitute $5$for $x$in the original equation.

$\begin{array}{rcl}{\log }_x\left(125\right)& =& 3\\ {\log }_5\left(125\right)& =& 3\\ 5^3& =& 125\\ 125& =& 125\end{array}$

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