Tuesday, September 20, 2016

calculus - Logarithm defined using the definite integral without Fundamental Theorem

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.




  1. log(xy)=log(x)+log(y)

  2. 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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