Working on an assignment but I've run into a stumbling block! I've got a couple of problems that I don't know how to do! The problems are attempting to have you define the $\log$ and $\exp$ functions. Thanks in advance! I can't wait to be done with this Analysis course!
Things I have already proven:
For any $ x \in (0,\infty)$, define $L(x)=\int_{1}^x {1\over t} dt$
$L(1/x)=-L(x)$
$L(x)$ is invertible and its inverse is $E(x)$
$E'(x)=E(x)$
$L(ax)=L(x)+L(a)$
$E(y+z)=E(y)E(x)$
Part A: Done
Let $n$ be a positive integer. Prove by induction that $E(nx)=E(x)^n$.
Part B: Done
Deduce from (a) that we also have $E(-nx)=E(x)^{-n}$, so (a) holds for all integers $n$.
Part C: My Problem Child
Deduce (give details) that for any rational number $r={m\over n}, $with $ n, m $ integers, $ m>0$, that $E(rx)=E(x)^r$
For this last part, I feel like I can just say something like let $r=n$, then the truth of Part A applies Part C, which implies that: $E(rx)=E(x)^r$, but that seems way to easy!
Thanks for the help!