The problem is to show

$$\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$$

for $t \gt 0$.

I'm pretty stuck. I thought about integration by parts and couldn't get anywhere with the integrand in its current form. I tried a substitution $u=e^{-x}$ and came to a new integral (hopefully after no mistakes)

$$ \int_0^1 \frac{u^{t-1}-1}{\log(u)}du, $$

but this doesn't seem to help either. I hope I could have a hint in the right direction... I really want to solve most of it by myself.

Thanks a lot!