Question

In the following exercises, solve for x.

$3{\log }_{5}x={\log }_{5}216$

Short answer

The value ofxfor the equation$3{\log }_{5}x={\log }_{5}216$ is$6$

Step-by-step solution

Step 1. Given data 

The given Logarithmic Equation is$3{\log }_{5}x={\log }_{5}216$

Step 2. Apply the power properties of the logarithm 

Use power properties of the logarithm${\log }_a\left(m^p\right)=p{\log }_a\left(m\right)$

$$\begin{gathered} 3{\log }_{5}x={\log }_{5}216 \\ {\log }_5{x}^3={\log }_{5}216 \end{gathered}$$

Step 3. Apply one to one property of logarithm

Use one to one property of the logarithm, if ${\log }_a\left(m\right)={\log }_a\left(n\right)$

then$m=n$

so

$$\begin{gathered} {\log }_5{x}^3={\log }_{5}216 \\ {x}^3=216 \\ x=\pm 6 \end{gathered}$$

the logarithm of a negative number is not possible

so$x=6$

Step 4. verification

Substitute $6$for $x$in the equation

localid="1645036653424" $$\begin{gathered} 3{\log }_{5}x={\log }_{5}216 \\ 3{\log }_{5}6\stackrel{?}{=}{\log }_{5}216 \\ {\log }_5{6}^3\stackrel{?}{=}{\log }_{5}216 \\ {\log }_{5}216={\log }_{5}216 \end{gathered}$$