how to generally approach these types of problems? $4^{x-1} + 4^{2-x} = 5 $

Question

$4^{x-1} + 4^{2-x} = 5 $

I know the result easily but I lack the general reasoning behind it? Should I use $ln$ or other approaches?

Answer 1

You have to turn it into an algebraic equation. This is suggested by $4^x$ and $4^{-x}$. Namely, if you call $4^x=t$ then you obtain an equation of the type $at^2+bt+c=0$.

Answer 2

Substitute $u=4^{x-1}$ so that ${1\over u}=4^{1-x}$. Then the equation becomes $$u+{4\over u}=5$$ or $$u^2-5u+4=0$$