What is the rule of equating exponents called? For example:

$$2^{2n-1} = 2^{n+2}
\Rightarrow 2n - 1 - n - 2 = 0 \Rightarrow n = 3$$

I couldn't find this rule in properties of exponents i.e when the bases are equal, the exponents can be equated. What is this rule called?
In the end, it comes down to the fact that the function
$$x\longmapsto 2^x$$
is "one-to-one"; that is, that different inputs yield different outputs. That is, that the graph of $y=2^x$ passes the so-called "horizontal line test": a horizontal line intersects the graph at most once.

It's the same reason that we can go from $a^3 = b^3$ to $a=b$: because the function $x\longmapsto x^3$ is one-to-one; and why we cannot go from $a^2=b^2$ to $a=b$: because $x\longmapsto x^2$ is not one-to-one (different inputs may give the same output; e.g., $(-1)^2 = 1^2$ even though $-1\neq 1$).

When a function is one-to-one, it has an inverse; and applying the inverse "undoes" what the original function does. That's what taking "logarithm base 2" is: the inverse of the exponential base 2.
If $f(x)$ is monotone (a slightly more restrictive condition than $1-1$ but satisfied by $f(x) = 2^x$) and $f(y) = f(x)$,  then $y = x$.

Most of the tables of "rules of exponents" I found were given names like "Rule 1" or "Rule 2", so I see the complaints about failing on giving a "name" somewhat arbitrary. If you wanted to use a "rule of exponents", then try combining the rule that might be called: "ratios imply subtraction of powers" and the "rule" that says if 2^x=1 that x=0 :

 a^m/a^n = a^(m-n) combined with the "rule" that if 2^m= 1 then m=0


You can then divide both sides 2n-1 and simplify.