Find $24 \cot^2 x$ 
Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24 \sin x} (24 \cos x) = \frac{3}{2}$. Find $24 \cot^2 x$.

From the given equation we have $24 \cos{x} = (24 \sin{x})^{\frac{3}{2}}$ and so $24\cot^2{x} = 24^2\sin{x}$. How do we continue?

You have $(24\cos x)^2 =(24\sin x)^3,\;$ or $\cos^2 x =24\sin^3 x.\;$
With $y=\sin x$ this gives the cubic $24y^3=1-y^2$ with the real solution
$y_1 = \frac{1}{3} \;$ or 
$$x=\arcsin\left(\frac{1}{3}\right) \approx  0.3398369$$
