How many real numbers $x$ satisfy the equation $$\left(|x^2-12x+20|^{\log{x^2}}\right)^{-1+\log x}=|x^2-12x+20|^{1+\log{\frac{1}{x}}}$$
I'm not sure how to deal with the absolute value in the question above. I can count 3 solutions; One is $2$ from the root of $x^2-12x+20$, and both solutions of $x^2-12x+19$.
Yet, I am told there are 6 real number solutions to the above. How can I deal with this problem?
Assuming tht your logarithm is the logarithm in base $10$, then forget the roots of $x^2-12x+20$; your equality becomes meaningless then. So, you have in fact four solutions: the roots of $x^2-12x+19$, which you have mentioned, plus the roots of $x^2-12x+21$. Also, $\frac1{\sqrt{10}}$ is a solution, since then $-1+\log(x)=-\frac32$, $\log(x^2)=-1$ and $1+\log\left(\frac1x\right)=\frac32$.