I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result for $q$:
$\displaystyle q = \frac{\log(3^p)}{\log(4)} + \frac{2\pi i c_{1}}{\log(4)}$
Nothing is said about what is $c_{1}$. Arbitrary constant? If $c_{1} \neq 0$ the result could be complex-valued I guess. I tried some calculations using the formula but the results do not make sense.
On
So to find all solutions, real and complex consider the definition of an exponential. $a^b=e^{b\log a}$. Also helps to mention that the complex logarithm of a complex number $r e^{it}$ is $\log\mid r \mid+2n\pi i$ with $n$ an integer. It's to do with the fact that the logarithm function is the inverse of a function that isn't one to one so you've to choose a 'branch'. Anyway...
Now write as $ e^{p\log 3} = e^{4\log q} $.
Take the complex log of both sides so that
$p\log 3 + 2n\pi i=q\log 4$ where $n$ is an integer and then divide across
Other's have discussed the output given by Wolfram|Alpha, so I'll provide only a hint for a solution.
Hint: For every $p \in \mathbb N$ with $p\geq 1$, we have that $4^p$ is even but $3^p$ is odd. A natural number cannot be both even and odd, so...