Find f(t) such that:
$\log (x-1)=\int_1^{\infty } \log \left(1-f(t)^{-x}\right) \, dt$
I am not familiar with solving integral equations. I was thinking of expressing logarithm inside integral with series, then moving integral inside this sum, but this leads me nowhere.
Do you have any idea? I do not expect there should be some nice formula for f(t) but at least some numerical method.