User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Exponent Combination Laws
I realized that the proofs of the Laws of Logarithms don't work according to the 2nd proof of Derivative of Natural Logarithm Function. Since I'm the one who added the second proof after the laws of logarithms were put up, I'm trying to add a proof for the laws that aren't circular. (While Googling it I found someone who encountered the same problem on a homework assignment, and (lucky for me?) he didn't post his resolution). To this end, can anyone tell me if $n$ is allowed to be an integer for this definition of $\exp x$ to be valid? If so, it would help me out, I think.


 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$

I might wait until next beginning of next year when I do sequences and series and limits at infinity in detail. The fact that I don't know the answer to this question is a good hint that I should wait :). At the very least, I think I know enough to prove laws of exponents for integer indices, should be a simple proof by induction, but I don't want to clutter PW with extraneous proofs. --GFauxPas 08:49, 22 December 2011 (CST)