Solve using auxiliary variable

Question

The question asks for me to solve the equation below using an auxiliary variable.

I don't know how to build the auxiliary variable for this problem, can anyone please help? Thanks.

$$10^{\log ( \log x )}-10^{\log (16/\log x)}=6$$

Answer 1

$$ 10^{\log ( \log x )}-10^{\log (16/\log x)}=6\Rightarrow 10^{\log ( \log x )}-10^{\log 16}10^{-\log(\log x)}=6\ . $$ Then set $10^{\log(\log x)}=t$, so your equation becomes $$ t-10^{\log 16}\frac{1}{t}=6\Rightarrow t^2-6t-10^{\log 16}=0\Rightarrow t=-2\text{ or }t=8\ . $$ We need to exclude $t=-2$, so the equation to be solved is $$ 10^{\log(\log x)}=8\Rightarrow 10^{\log(\log x)}=10^{\log 8}\Rightarrow \log(\log x)=\log 8\Rightarrow \log x=8\Rightarrow \boxed{x=10^8}\ . $$