This question is similar to this question, Relation between elements with fixed exponent over different $\mathbb{Z}^\times_p$
For each prime $p$ that has a primitive root $3$ and for all $a\in\mathbb{Z}^+$, there exists a $b\in\mathbb{Z}^+$ such that $2^a \equiv 3^b \pmod{p}$. Is there any relation between $a$ and $b$ that can be expressed without $p$? If not then is there one for the primes with a primitive root $5$?
