Of course there is no unambiguously 'right' choice between two or more definitions that give the same answer, but I think that it is more common to switch your (1) and (2), and define ln(x) as \int_1^x dx/x.
Re:A Quirky Function
by GFK_of_xmaspast on Saturday June 25, 2016 @11:46AM
(#11976704)
What's wrong with defining exp(z) in terms of the usual taylor series?
Yeah I guess that works as well.
I wonder though how hard it would be to prove basic equalities (e.g. e^(x + y) = e^x * e^y)... Probably not too hard since that's how you define the complex exponential function.
You can define a^x or 0^x by its algebraic properties only for rational numbers (i.e. fractions). In algebra, it makes sense that 0^0 = 1, but that won't allow you to plot 0^x as a function.
To plot it as a function, you need to define it for all real numbers, which, AFAIK, you need to do analytically:
1. First, define e^x as the limit (1 + x/n)^n as n goes to infinity (n is a natural number, so this definition uses the standard algebraic exponentiation).
2. Then, you define ln(x) as the reverse of e^x (after proving that e^x is monotonous and continuous).
3. Then, define a^x as e^(x * ln a).
4. Then, 0^x is defined as a^x as a -> 0, which is 0 for x > 0. In the same way, you can see that x^0 is 1 for x > 0. Therefore, there is no "correct" (analytically) result for 0^0 (i.e. such that would make the function f(x, y) = x ^ y continuous at (x, y) = (0, 0)).