How do I prove that a certain price is price of European option in Black-Scholes framework

I want to show whether the following price at t is of a european option in Black-Scholes Framework. $$S_tlog_e (S_t^3)$$ Is it just trying to substitute the function (and partial derivates) in the Black-Scholes PDE?

Yes it is actually just substituting it into the Black Scholes PDE. If the PDE is satisfied, $$V(t,S(t)),t\ge 0$$ is a martingale and hence $$V(t,S(t)) = E_t (V(T,S(T))$$ so that $$V(t,S(t))$$ is the expected value, at time t, of an option that pays $$V(T,S(T))$$ at time $$T$$. Here I assumed $$r=0$$ for simplicity

If it was a solution to a European option PDE pricing problem, at time $$T$$ you have $$V(T,S) = S \log{S^3}$$.

Based on this terminal function, you can then solve the BS PDE using a semi-analytical approach to find $$V(t,S)$$. My guess is it will not be the candidate you posted (haven't checked).