Convergence of improper integral with logarithm

Question

I would like to determine the nature of $A$ without calculating it. $$ A= \int_0^1 \ln(1-t^{a}) dt . $$

In $t=1$ we have a problem, so how should I proceed?

Answer 1

Let $u=1-t$ then

$$\ln(1-t^a)=\ln(1-(1-u)^a)=_0\ln(1-1+au+o(u))\sim_0\ln(au)$$ and since the integral $\int_0^1\ln xdx $ is convergent then the given integral is also convergent for all $a>0.$