Is $a^n$ an exponent of $a$? I see that definition says that $n$ is an exponent. But, the name of the function is normally expanded to the results computed by the function. That is, if we raise $a$ into power $n$ we say that result is a power (or exponent) of $a$. For instance, you say a $\sin\pi$. You apply the term 'sine' to the value of the function. So, here both $n$ and $a^n$ must be exponents. Right? 
Identically, WP says that power is the same as exponent in the context of exponentiation. Are both n and $b^n$ powers of b?

It might be useful to list a number of related bits of terminology:
$a^b$ is a power of $a$.  In this expression, $b$ is the exponent, and $a$ is the base.  
If we think of $a$ as fixed and $b$ as varying, then the function sending $b$ to $a^b$ is an exponential function (with base $a$).
If we think of both $a$ and $b$ as varying, the function sending $a,b$ to $a^b$ is exponentiation.
