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How to prove the double integral definition of logarithm?

  1. Sep 20, 2015 #1


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    Where does this definition come from: $$\ln n = \int_{0}^{\infty} \int_{1}^{n} e^{-xt} dx dt$$
    Thank you very much.
  2. jcsd
  3. Sep 20, 2015 #2


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    Staff: Mentor

    Exchange the order of the integrals, and you can calculate them easily.
  4. Sep 20, 2015 #3


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    I don't know but it seems follows from elementary integrals, if you change the order (verify you can do it!) the first is ##\int_{0}^{+\infty}e^{-xt}d\,t=\frac{1}{x}## ( it is improper but converge ) and what remains is simply ##\int_{1}^{n}\frac{1}{x}d\,x## that is ##\log{n}##...
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