Step-by-Step Solution
Key Concept: Using logarithmic properties to convert exponential equations into trigonometric identities involving tangent
In the interval $[-\pi, \pi]$, all values of $\sin x$ are positive as well as $|\cos x|$. Taking logarithm of both sides: $\sin x \log 2 = \cos x \log 3$. Since $\tan x = \frac{\log 3}{\log 2}$, and the value of $\tan x$ is positive in the 1st and 3rd quadrants and negative in the 2nd and 4th quadrants, $\tan x$ will repeat its value in all four quadrants. Therefore, the number of solutions is $4$.
Correct Answer: 4