# How to prove the double integral definition of logarithm?

1. Sep 20, 2015

### td21

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

2. Sep 20, 2015

### Staff: Mentor

Exchange the order of the integrals, and you can calculate them easily.

3. Sep 20, 2015

### Ssnow

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 $\log{n}$...