How to prove the double integral definition of logarithm?

  • Thread starter td21
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  • #1
td21
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Main Question or Discussion Point

Where does this definition come from: $$\ln n = \int_{0}^{\infty} \int_{1}^{n} e^{-xt} dx dt$$
Thank you very much.
 

Answers and Replies

  • #2
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Exchange the order of the integrals, and you can calculate them easily.
 
  • #3
Ssnow
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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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