If one wants to define the natural logarithm using (Riemann) integrals, he could do as follows:
log(x):=∫x11tdt
Let's assume we hadn't defined the exp-function yet. How can one prove some basic logarithmic identities WITHOUT using the Fundamental Theorem of Calculus.
- log(xy)=log(x)+log(y)
- log′(1)=1
Edit:
This exercise came up as an homework for a Calculus I class at my university. They explicitly stated not to use the exp-function or the FTC. Since I didn't know how they were supposed to solve this with these restrictions, I thought I post this problem here. (Note: This homework was due a few weeks ago. So posting the solution here shouldn't be a problem.)
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