Inverse Laplace transform of $\log(s)$

Question

I would like to calculate the ILT of the function $\log\left(s\right)$. I don't know if my calculations are right. Since $$F(s)=\log\left(s\right),\,\textrm{Re}(s)>0$$ then $$F^{\prime}\left(s\right)=\frac{1}{s}$$ so if we put $$f\left(t\right)=L^{-1}\left(\log\left(s\right)\right)\left(t\right)$$ we have, using the properties of the Laplace transform, that $$L\left(tf\left(t\right)\right)\left(s\right)=L\left(u\left(t\right)\right)\left(s\right)$$ where $u(t)$ is the unit step function. So $$f\left(t\right)=\frac{u\left(t\right)}{t}.$$ Are my calculations correct? Thank you.