I recently joined an econ class. I am so lost on how to prove their equality. As a math standpoint, these are completely different equations. Please help!
u1 (x1, x2) = x1^(2/3) x2^(1/3) u2 (x1, x2) = 4ln (x1) + 2 ln(x2) +3
According to the logical positivist view of decision theory, utility functions are just descriptions of observable behavior and have no intrinsic meaning absent this vantage. In other words, $u(x) > u(y)$ indicates that $x$ would be chosen in favor of $y$, but the magnitude of the difference carries no addition information. If $u(x) = 1$ and $u(y) = 0$ the decision maker prefers $x$; if $u(x) = 10000$ and $u(y)= 0$ the decision maker prefers $x$. In either case, we can draw the same conclusions.
What does this mean with regard to your question? It tells us what we mean by equality of utility functions: we mean that these two different (so no definitionally equal) functions represent the same observable data. Specifically, exactly when $u_1(x) > u_1(y)$ do we also have $u_2(x) > u_2(y)$. As mentioned in the comments, this is precisely when we can compose $u_1$ with a strictly increasing function and pop out $u_2$. (This is because all we care about is the ordering between objects, and strictly increasing functions preserve order).
So, what might our strictly increasing function look like? Well, we can take the log, so that we get $\frac23 ln(x_1) + \frac13 ln(x_2)$ then multiplying by 6 and adding 3 does the trick. Each of these was strictly increasing so too is the composition: $$ (x,y) \mapsto 6 (ln(x) + ln(y)) + 3$$ is our desired function.