# real analysis – prove \$log x\$ is integrable near the origin

Prove that $$log(x) in L^1((0,1))$$.which has been shown in this post.

The first idea is using the integral $$int_epsilon^1 log(x) dx$$ to approximate it.which may be the definition in classical analysis.I was bit confused for why this limit: $$lim_{epsilon to 0} I(epsilon) = int_0^1 log x dx$$ holds in Lebesgue integral. Maybe we need to take an absolute value first $$int_epsilon^1 |log(x)|dx$$ first then using the Monotone convergence theorem for non-negative function, rest of the proof are the similar ,is my interpretation correct?

The second idea also shows in this post:using the fact that $$x^{-alpha} in L^1((0,1))$$ for $$alpha <1$$ then near the origin ,we have $$x^{alpha}log x to 0$$ for all $$alpha >0$$ which means exist a small neiborhood near origin says $$(0,delta)$$ such that if $$xin (0,delta)$$ we have $$|x^alog x|le 1/2$$ i.e. $$|log x|le frac{1}{2}x^{-a}$$ since the RHS is integrable hence log is also integrable near origin,is my interpretation correct?