What's the inverse operation of exponents?

Addition and multiplication are commutative, so there is just one inverse function.

Exponents are not commutative; $2^8 \not= 8^2$. So we need two different inverse functions.

Given $b^e = r$, we have the "$n$th root" operation, $b = \sqrt[e] r$. It turns out that this can actually be written as an exponent itself: $\sqrt[e] r = r^{1/e}$.

Again, given $b^e = r$, we have $e = \log_b r$, the "base-$b$ logarithm of $r$".

These functions are the logarithms, and they are fundamentally important. For $a = b^c$ (where $b > 0$) we write: $$c = \log_b a,$$ which we can take to be the definition of $\log_b$. We read the operation as "logarithm, base $b$," or "base $b$ logarithm".

In particular, we have $$\log_a (a^b) = b \qquad\text{and}\qquad a^{\log_a b} = b.$$ Of special interest is the natural logarithm, denoted by $\ln$ or $\log$, the logarithm of base $e$. (NB that sometimes $\log$ can also denote base $10$, or base $2$, depending on context.)

Logarithmic identities correspond to exponential identities. From example, from the definition we can conclude that $$\log_b (pq) = \log_b p + \log_b q$$ (for $p, q > 0$), which corresponds to the identity $b^{pq} = b^p b^q$.

Perhaps counterintuitively, sometimes it is convenient to define the natural logarithm first and then define the exponential function $x \mapsto e^x$ to be its inverse, which leads to the slightly antiquated name antilog for an exponential function $x \mapsto b^x$.

Edit Some of the other answers here pointed out quite rightly that one can also ask about the inverse of functions where the variable is in the base, i.e., functions $x \mapsto x^a$, and inverses of these functions$^*$ (at least when $a > 0$) are just $x \mapsto x^{1/a}$, which we often write as $x \mapsto \sqrt[a]{x}$. These functions are called power functions (note that the inverse of a power function is again a power function), and we reserve the name exponential function for functions $x \mapsto b^x$ where the variable is in the exponent, i.e., those to which the logarithms are inverses.

$^*$For some $a$ (in particular, even integers), we need to restrict the map $x \mapsto x^a$ to $[0, \infty)$ in order to take an inverse.

There are two inverse operations of exponentiation.

Logarithm

$$ \log _{b} a $$

It's read "base-$b$ logarithm of $a$". And it means "the exponent which $b$ must be raised to, so that the result is $a$".

Root

$$ \sqrt[b] a $$

It's read "$b$-th root of $a$". And it means "the number which, when raised to $b$, produces $a$".