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 $2$?
Does $\exp(f(x)) = e^{f(x)}$
exponentiationnotation
Related Solutions
The symbol $\therefore$ is used in the book An introduction to formal logic, by Peter Smith, a very nice book indeed. It is a symbol of the formal language of logic, used as inference marker, to signal that an inference is being drawn from the premises. I quote from p. 110 (PL is the language of propositional logic, later in the book extended to full first-order theory logic):
Rather oddly, counting $\therefore$ as part of PL is not standard; but we are going to be very mildly deviant. However, our policy fits with our general view of the two-stage strategy for testing inferences. We want to be able to translate vernacular arguments into a language in which we can still express arguments; and that surely means not only that the individual wffs of PL should be potentially contentful, but also that we should have a way of stringing wffs together which signals that an inference is being made
Then, on p. 119, there is a nice discussion about the difference between $\vDash$ (a metalinguistic symbol, which does not belong to PL, for talking about the relationship of some wffs in PL) and $\therefore$ (a symbol of PL)
The confusion arises because there are two inconsistent, albeit related, uses of the word function. The older use is what may be called a dependent variable, in (for example) the phrase “$y$ is a function of $x$” or, more specifically, “the function $y=2x+4$”. This usage is still common among non-mathematicians who employ mathematics. This language tends to be avoided by present-day mathematicians, because it implies, in this case for example, that a function is a kind of real number (which depends on another, freely specifiable, real number). The function here is not $y$ but (in simple terms) the rule that specifies how $y$ is obtained from $x$. In the modern sense, a function can be precisely defined as a kind of mathematical object, which is quite distinct from the values (e.g. $y$) associated with the function.
Best Answer
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.
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 $e$.