Does $\exp(f(x)) = e^{f(x)}$?

Question

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 $2$?

Answer 1

Is this correct, or do these mean something different?

Yes, it is correct.

And yes, it is different — typographically. If the exponent gets complicated, typesetting the exponent can result in tiny symbols that might be hard to read. Using $\exp$ makes the exponent "one level" bigger. Using notation $e^x$ is preferred, IMHO, for simple exponents because it is shorter and needs less parenthesis.

Also, in computer science, should these be replaced rather with $2^{f(x)}$ since the base mostly considered is $2$?

No, of course not. Some math libs provide functions like $\operatorname{exp2}$ and $\operatorname{exp10}$ for bases 2 and 10, respectively, but $\exp$ is still base $e$.

Answer 2

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

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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.