Problem: Let $x, y$ and $z$ all exceed 1 and let $w$ be a positive number such that $\log _x w=24, \log _y w=40$ and $\log _{x y z} w=12$. Find $\log _z w$ Solution: Parsing definitions, $x^{24} = y^{40} = (xyz)^{12} = w$. We can simply use these equations to reduce unwanted terms in $x$ and $y$ starting with squaring both sides: $\begin{align*} (xyz)^{24} & = w^2 \\ \iff x^{24}y^{24}z^{24} &=w^2 \\ \iff y^{24}z^{24} &= w \\ \iff y^{40}z^{40} &= w^{\frac{5}{3}} \\ \iff z^{40}& =w^{\frac{2}{3}}\\ \iff z^{60}& =w \\ \iff \log_zw &=60\end{align*}$ No tricks; just unpacking definitions and working the algebra.