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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:

Let n be a positive integer. Prove by induction that $E(nx)=E(x)^n$.

Part B:

Deduce from (a) that we also have $E(-nx)=E(x)^{-n}$, so (a) holds for all integers n.

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    Do you know what the structure of a proof by induction is? Did you check the base case, and try to set up the inductive step? I suggest using $(n+1)x=nx + x$ and $ 0 = nx - nx$ for parts A and B, respectively.2012-12-14
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    Hint: Part A: $E((n-1)x+x) = E((n-1)x)E(x)$ Part B: Prove E(0) = 1 and then from part A, what is $E(nx+ (-nx))$?2012-12-14
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    Please do not delete you question. This posting could be helpful to someone else in future2012-12-17

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