How to figure of the Laplace transform for $\log x$?

The Laplace transform of the power function is:

$$ \int_0^\infty e^{-st} t^a dt = \frac{\Gamma(a+1)}{s^{a+1}} $$

Differentiate with respect to $a$ using differentiation under the integral sign:

$$ \int_0^\infty e^{-st} t^a \log{t} dt = \frac{\Gamma'(a+1) s^{a+1} - \Gamma(a+1) s^{a+1} \log{s}}{(s^{a+1})^2} = \frac{\Gamma'(a+1) - \Gamma(a+1) \log{s}}{s^{a+1}} $$

Now plug in $a = 0$ to get what you want.