trigonometry - Find domain of function $logleft(cosleft(log xright)right)$

Find domain of function $f(x)=\log\left(\cos\left(\log x\right)\right)$

At the beginning $x>0$
but I have no idea how to handle logarithm
$\cos\left(\log x\right)>0\\\log x>\arccos 0=\frac{\pi}{2}\\\log x>\log 10^{\frac{\pi}{2}}\Rightarrow x>10^{\frac{\pi}{2}}$
I think I'm doing something wrong

Answer

We need

• $x>0$




• $\cos(\log x)>0$

$$\implies 0\le \log x < \frac \pi 2 \cup -\frac \pi 2 +2n\pi < \log x <\frac \pi 2 +2n\pi$$