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 $\log_{xyz} w = 12$. Find $\log_z w$.
The [logarithmic](https://artofproblemsolving.com/wiki/index.php/Logarithm) notation doesn't tell us much, so we'll first convert everything to the equivalent exponential forms.
$x^{24}=w$, $y^{40}=w$, and $(xyz)^{12}=w$. If we now convert everything to a power of $120$, it will be easy to isolate $z$ and $w$.
$x^{120}=w^5$, $y^{120}=w^3$, and $(xyz)^{120}=w^{10}$.
With some substitution, we get $w^5w^3z^{120}=w^{10}$ and $\log_zw=\boxed{060}$.