Question

Solve the equation.

${\log }_x\left(\frac{1}{8}\right)=3$

Short answer

The solution to the equation, ${\log }_x\left(\frac{1}{8}\right)=3$is, $x=\frac{1}{2}$.

Step-by-step solution

Step 1. Given Information

The given equation to be solved is ${\log }_x\left(\frac{1}{8}\right)=3$.

Step 2. Formula used

Rule of change from logarithm to exponents:

${\log }_{a}b=y\Rightarrow a^y=b$

Step 3. Finding Solution

The given equation is ${\log }_x\left(\frac{1}{8}\right)=3$.

Use rule of change from logarithm to exponents:

$$\begin{gathered} {\log }_x\left(\frac{1}{8}\right)=3 \\ {x}^3=\frac{1}{8} \\ (x{)}^3={\left(\frac{1}{2}\right)}^3 \\ x=\frac{1}{2} \end{gathered}$$

The solution to the given equationis,$x=\frac{1}{2}$.