College Algebra, Chapter 5, 5.4, Section 5.4, Problem 56

Find the value of $x$ to make the statement $(\log x)^3 = 3 \log x$

$
\begin{equation}
\begin{aligned}
(\log x)^3 &= 3 \log x\\
\\
(\log x)(\log x)(\log x) &= 3 \log x && \text{Expand } (\log x)^3\\
\\
\frac{\left( \cancel{\log x} \right) (\log x) (\log x)}{\cancel{\log x}} &= \frac{3 \cancel{\log x}}{\cancel{\log x }} && \text{Divide by } \log x\\
\\
(\log x)(\log x) &= 3 && \text{Cancel out like terms}\\
\\
(\log x)^2 &= 3 && \text{Simplify}\\
\\
\log x &= \sqrt{3} && \text{Take the square root of each side}\\
\\
10^{\log x} &= 10^{\sqrt{3}} && \text{Raise 10 to each side}\\
\\
x &= 10^{\sqrt{3}}
\end{aligned}
\end{equation}
$

Then by checking,

$
\begin{equation}
\begin{aligned}
\left(\log 10^{\sqrt{3}}\right)^3 &= 3 \log 10^{\sqrt{3}}\\
\\
5.1962 &= 5.1962
\end{aligned}
\end{equation}
$