What is process/function to cancel base (in value with exponential)?

You use the logarithm of the given base. For instance, if you have (in the natural base)

$$e^{f(x)}$$ you can take the natural logarithm of that and get

$$\ln\left(e^{f(x)}\right)=f(x)\ln(e)=f(x)$$

So in your equation, you take the $\log_3$ on both sides to obtain $$\log_3\left(3^{3x}\right)=\log_3\left(3^{2y+1}\right) \leftrightarrow 3x\log_3(3)=(2y+1)\log_3(3) \leftrightarrow$$

$$3x=2y+1$$

The function $f(t)=3^t$ is increasing. So if $f(a)=f(b)$ we have $a=b$.

see for basic understanding i can say you that if bases are same and two numbers are in equality then there powers must be equal

$$2^z=2^{33}$$ this means $$z=33$$ but if it is given like $$2^g=15.(2)^{33}$$ then you cannot write $g=33$

for proving this we have to use logarithms.