What is the rule of equating exponents called?

Question

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?

Answer 1

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.

Answer 2

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 $2n-1=n+2$.