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?
 A: 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.
A: It's merely the fact that exponential functions whose base is a positive number other than 1 are one-to-one functions.  If $f$ is a one-to-one function and $f(2n-1) = f(n+2)$ then $2n-1=n+2$.
A: 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.
