Question

In the following exercises,solve each logarithmic equation.

${\log }_{a}24=3$

Short answer

The solution$a=\sqrt[3]{24}$.

Step-by-step solution

Step 1.Given information

Consider the logarithmic equation ${\log }_{a}24=3$.

The logarithmic from$y={\log }_{a}x$ is equivalent to exponential form$x=a^y$.

Step 2.Find the exponential form 

Comparing ${\log }_{a}24=3$with the standard form $y={\log }_{a}x$,you have

$y=3,a=a,x=24$

Convert logarithmic equation in exponential equation as shown below

$$\begin{gathered} x=a^y \\ 24=a^3 \\ {a}^3=24 \\ So,theequationinexponentialformis{a}^3=24 \end{gathered}$$

Step 3.Solve the equation and find the value of a.

$$\begin{gathered} {\left(a^3\right)}^{\frac{1}{3}}={\left(24\right)}^{\frac{1}{3}}[Raiseradical\frac{1}{3}tothebothsides] \\ a=\sqrt[3]{24} \\ So,thesolutionisa=\sqrt[3]{24} \end{gathered}$$

Step4.Check

To check the solution,replace a by $\sqrt[3]{24}$ in original equation

$$\begin{gathered} {\log }_{a}24=3 \\ {\log }_{\sqrt[3]{24}}24=3[a=\sqrt[3]{24}] \\ {\left(\sqrt[3]{24}\right)}^3=24 \\ 24=24\left[true\right] \\ Therefore,thesolutionisa=\sqrt[3]{24} \end{gathered}$$