How do you multiply these logarithms and find the domain?

Question

I'm supposed to solve for $x$ and find the domain. I know that adding two logs together is multiplication of the numbers, but what if two logs are completely multiplying each other? $$\log_{10}(0.1x^2)\log_{10}x=1$$

Answer 1

The domain of the LHS of the original equation is $x\in(0,\infty)$, because of the presence of $\log_{10}x$.

By logarithm rules (justified by the domain of $x$), we can take the multiple of 0.1 and the square out of $0.1x^2$: $$\log_{10}(0.1x^2)\log_{10}x=(\log_{10}0.1+2\log_{10}x)\log_{10}x=(2\log_{10}x-1)\log_{10}x=1$$ We denote $\log_{10}x=y$: $$(2y-1)y=2y^2-y=1$$ $$2y^2-y-1=0$$ Solving for $y$, we get $y=1$ and $y=-\frac12$, corresponding to solutions of $x=10$ and $x=\frac1{\sqrt{10}}=0.316\dots$