Let $a>0$, Show that $\int_{0}^{a} \ln(x) dx $ converges

Question

so when I calculated the improper integral I got, $$\int_{0}^{a} \ln(x) dx =[a\ln(a)]-a~,\qquad a>0$$ substituting as the limit goes from $a$ to $\infty$ I believe the integral diverges? I'm not sure what I may be doing wrong.