Show $\frac{1}{\log(x)}$ is in $L^p ((0,0.5))$ for $p \geq 1$

Question

Show $\frac{1}{\log(x)}$ is in $L^p ( (0,0.5))$ for $p \geq 1$ without using the $\operatorname{Li}(x)$ function or using $u = \log(x)$ substitution (that approach gives an infinite series and is not good for evaluating $x=0$).

Answer 1

$f\colon x\mapsto\frac1{\lvert\log x\rvert}$ is continuous on $(0,\frac12)$ and approach a finite limit on either end. So $f$ is uniformly continuous hence $f^p$ is integrable (and the case $p=\infty$ is immediate from uniform continuity of $f$).