Question

What is the range of f(x)=ln(cosx)? Why? Please do not use a calculator.

Answer 1

$The\ domain\ of\ \ln(x)\\D:x \ \textgreater \ 0\\\\\cos x\in[-1;\ 1]\\\\The\ domain\ of\ \ln(\cos x)\\D:\cos x \ \textgreater \ 0\to \cos x\in(0;\ 1]\to x\in\left(-\dfrac{\pi}{2}+2k\pi;\ \dfrac{\pi}{2}+2k\pi\right);\ k\in\mathbb{Z}\\\\therefore\\\\range\ of\ \ln x\ where\ x\in(0;\ 1]\ is\ equal\ (-\infty;\ 0]\\\\Answer:\boxed{R:y\in(-\infty;\ 0]}$