Comment by lutusp

> The use of ln 2 for argument range reduction has nothing to do with half lives.

That is a false statement.

> In fact, only doubling or halving times are directly measured for radioactive decay or population growth.

That is a false statement -- in population studies, as just one example, the logistic function (https://en.wikipedia.org/wiki/Logistic_function) tracks the effect of population growth over time as environmental limits take hold. This is a detailed model that forms a cornerstone of population environmental studies. To be valid, it absolutely requires the presence of e^x in one or another form.

> ... because the numbers are represented as binary numbers in computers and the functions are evaluated with approximation formulae that are valid only for a small range of input arguments.

That is a spectacularly false statement.

> There is no difference in the definition set between 2^x and e^x.

That is absolutely false, and trivially so.

> No, you did not try to understand what I have written.

On the contrary, I understood it perfectly. From a mathematical standpoint, 2^x cannot substitute for e^x, anywhere, ever. They're not interchangeable.

I hope no math students read this conversation and acquire a distorted idea of the very important role played by Euler's number in many applied mathematical fields.