The number "e" itself is never needed in any application.
It is not used for computing the value of ln(2) or of log2(e), which are computed directly as limits of some convergent series.
As I have said, there is no reason whatsoever for knowing the value of e.
Moreover, there is almost never a good choice to use the exponential function or the hyperbolic logarithm function (a.k.a. natural logarithm, but it does not really deserve the name "natural").
For any numeric computations, it is preferably to use everywhere the exponential 2^x and the binary logarithm. With this choice, the constant ln 2 or its inverse appears in formulae that compute derivatives or integrals.
People are brainwashed in school into using the exponential e^x and the hyperbolic logarithm, because this choice was more convenient for symbolic computations done with pen on paper, like in the 19th century.
In reality, choosing to have the proportionality factor in the derivative formula as "1" instead of "ln 2" is a bad choice. The reason is that removing the constant from the derivative formula does not make it disappear, but it moves it into the evaluation of the function and in any application much more evaluations of the functions must be done than computations of derivative or integral formulae.
The only case when using e^x may bring simplifications is in symbolic computations with complex exponentials and complex logarithms, which may be needed in the development of mathematical models for some linear systems that can be described by linear systems of ordinary differential equations or of linear equations with partial derivatives. Even then, after the symbolic computation produces a mathematical model suitable for numeric computations it is more efficient to convert all exponential or logarithmic functions to use only 2^x and binary logarithms.
From your other responses in this thread, it looks like you do concede that e is useful in symbolic computation, and others use the phraseology "how the function is implemented", which is quite a silly thing to say in a classical math context, but not in a computational context.
I didn't understand immediately that you were talking about using values related to e in a computational context. But your comment about "brainwashing" seems a bit off. Are you saying that programmers bring e and ln with them into code when more effective constants exist for the same end? That's probably true. But brainwashing is far too strong, since things need to be taught in the correct order in math in order for each next topic to make sense. e really only comes in when learning derivative rules where it's explained "e is a number where when used as the base in an exponential function, that function's derivative is itself." Math class makes no pretense that you ought to use any of it to inform how you write code, so the brainwashing accusation seems off to me.
It is not used for computing the value of ln(2) or of log2(e), which are computed directly as limits of some convergent series.
As I have said, there is no reason whatsoever for knowing the value of e.
Moreover, there is almost never a good choice to use the exponential function or the hyperbolic logarithm function (a.k.a. natural logarithm, but it does not really deserve the name "natural").
For any numeric computations, it is preferably to use everywhere the exponential 2^x and the binary logarithm. With this choice, the constant ln 2 or its inverse appears in formulae that compute derivatives or integrals.
People are brainwashed in school into using the exponential e^x and the hyperbolic logarithm, because this choice was more convenient for symbolic computations done with pen on paper, like in the 19th century.
In reality, choosing to have the proportionality factor in the derivative formula as "1" instead of "ln 2" is a bad choice. The reason is that removing the constant from the derivative formula does not make it disappear, but it moves it into the evaluation of the function and in any application much more evaluations of the functions must be done than computations of derivative or integral formulae.
The only case when using e^x may bring simplifications is in symbolic computations with complex exponentials and complex logarithms, which may be needed in the development of mathematical models for some linear systems that can be described by linear systems of ordinary differential equations or of linear equations with partial derivatives. Even then, after the symbolic computation produces a mathematical model suitable for numeric computations it is more efficient to convert all exponential or logarithmic functions to use only 2^x and binary logarithms.