Does $\exp(f(x)) = e^{f(x)}$? I've rarely seen the notation $\exp(f(x))$ but whenever I do I just replace it with $e^{f(x)}$. Is this correct, or do these mean something different? Also, in computer science, should these be replaced rather with $2^{f(x)}$ since the base mostly considered is $2$?

For typographical puroses, when the exponent is complicated, it is preferable to write it with $\exp$ for readability.  Compare
$$
\exp\left(\frac{(\ln x)^2}{\sqrt{x^2+1}}\right) = 
e^{\frac{(\ln x)^2}{\sqrt{x^2+1}}}
$$

Sometimes in elementary texts we define a function $\exp$, develop some of its properties, and only after a while (from the properties) can we see that it is reasonable to write $e^x$ for this function.
