\lim_{x\rightarrow 0}\frac{\ln(\cos x)}x=\lim_{x\rightarrow 0}\frac{\frac{\ln(\cos x)}{\cos x-1}\cdot(\cos x-1)}x=\left(\lim_{x\rightarrow 0}\frac{\ln(\cos x)}{\cos x-1}\right)\cdot\lim_{x\rightarrow 0}\frac{\cos x-1}{x}