Answer: 60

Small Hint:

Rewrite each given logarithm with base $w$

Big Hint:

Expand $\log_w(xyz)$ as a sum of three logarithms

Solution:

Taking reciprocals of the given logarithms gives $$\begin{aligned} \log_w x&=\frac1{24},\\ \log_w y&=\frac1{40},\\ \log_w(xyz)&=\frac1{12}. \end{aligned}$$ Therefore $$\log_w z=\frac1{12}-\frac1{24}-\frac1{40} =\frac1{60}.$$ Taking the reciprocal yields $\log_z w=60.$