Question

In the following exercises, solve each logarithmic equation.

$\log (x^2-21)=2$.

Short answer

The solution is$x=11,-11$.

Step-by-step solution

Step 1. Given and explanation. 

We have$\log (x^2-21)=2$.

We know that if there is an expression as${\log }_e\left(x\right)=y$, it means thatif we need to convert the logarithmic expression into exponential form.

Then simplifying the equation will give us the answer.

Step 2. Converting into exponential form. 

We have $\log (x^2-21)=2$.

This can also be written as ${\log }_{10}(x^2-21)=2$

Converting into exponential forms gives us,
$x^2-21={10}^2$

Step 3. Solving for x.

We have $x^2-21={10}^2$.

Simplifying it gives us,

$$\begin{gathered} x^2-21=100 \\ {x}^2=100+21 \\ {x}^2=121 \\ x=\pm 11 \end{gathered}$$

Thus we get the solution as$x=\pm 11$.

Step 4. Checking the solution.

We have $x=\pm 11$.

Substituting the values give us,

1. When $x=11$
   $$\begin{gathered} \log ({\left(11\right)}^2-21)=2 \\ \log (121-21)=2 \\ \log \left(100\right)=2 \\ {\left(10\right)}^2=2 \\ 2=2 \end{gathered}$$
   Hence verified.
2. When $x=-11$
   $$\begin{gathered} \log ({(-11)}^2-21)=2 \\ \log (121-21)=2 \\ \log \left(100\right)=2 \\ {\left(10\right)}^2=2 \\ 2=2 \end{gathered}$$
   Hence verified.