Question

In the following exercises,solve each logarithmic equation.

${\log }_{a}27=3$.

Short answer

a=3

Step-by-step solution

Step 1.Given information

Consider the logarithmic equation ${\log }_{a}27=3$

The logarithmic from${\log }_{a}27=3$ is equivalent to exponential form$x=a^y$.

Step 2.Solve the equation and find the value of a.

Comparing ${\log }_{a}27=3$with the standard form $y={\log }_{a}x$,you have

Convert logarithmic equation in exponential equation as shown below

$$\begin{gathered} x=a^y \\ 27=a^3 \\ {a}^3=27 \\ So,theequationinexponentialformis{a}^3=27. \end{gathered}$$

Now solve the equation as

role="math" localid="1644925625777" $$\begin{gathered} {\left(a^3\right)}^{\frac{1}{3}}={\left(27\right)}^{\frac{1}{3}}[RaiseRadical] \\ {\left(a^3\right)}^{\frac{1}{3}}={\left(3^3\right)}^{\frac{1}{3}} \\ {\left(a\right)}^{3\frac{1}{3}}={\left(3\right)}^{3\frac{1}{3}} \\ a=3 \\ So,thesolutionisa=3 \end{gathered}$$

To check the solution,replace a by 3 in original equation

$$\begin{gathered} {\log }_{a}27=3 \\ {\log }_{3}27=3[a=3] \\ {3}^3=27 \\ 27=27\left[true\right] \\ Therefore,thesolutionisa=3 \end{gathered}$$