Question

In the following exercises, solve for x.

${\log }_7\left(x\right)=-2$

Short answer

The solution is $x=\frac{1}{49}$.

Step-by-step solution

Step 1. Convert into exponential form

Consider the logarithm equation ${\log }_7\left(x\right)=-2$.

Its equivalent exponential form is given as

$\begin{array}{rcl}{7}^{-2}& =& x\\ \frac{1}{7^2}& =& x\\ \frac{1}{49}& =& x\end{array}$

So the solution is$\frac{1}{49}$.

Step 2. Check the solution

Substitute $\frac{1}{49}$for $x$in the original equation.

$\begin{array}{rcl}{\log }_7\left(x\right)& =& -2\\ {\log }_7\left(\frac{1}{49}\right)& =& -2\\ 7^{-2}& =& \frac{1}{49}\\ \frac{1}{7^2}& =& \frac{1}{49}\\ \frac{1}{49}& =& \frac{1}{49}\end{array}$

As the statement is true, so the found solution is correct.