# Why is the logarithm squared in this equation? $\log^2x-3\log x=\log x^2-4$

I just don't understand why the log is squared $$\log^2x-3\log x=\log x^2-4$$

• $(\log(x))^2$ is often represented as $\log^2(x)$. Notice that it is a quadratic equation in terms of $\log(x)$ Sep 16, 2020 at 5:49
• See also: What does $\log^{2}{x}$ mean? Sep 16, 2020 at 8:22

The notation stands for: $$\log^2 x=(\log x)(\log x)$$.
Then use that $$\log x^2=2\log x$$ and let $$\log x=t$$ to obtain a quadratic.
Let $$log x=y$$, then $$log^2 x-3log x=log x^2-4$$ $$y^2-3y=2y-4$$ $$y^2-5y+4=0$$ $$(y-1)(y-4)=0$$ $$y=1$$ or $$y=4$$ $$log x=1$$ or $$log x=4$$
Hence $$x=10$$ or $$x=10000$$