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,

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