Logarithmic limits problems and solutions

The list of limits problems which contain logarithmic functions are given here with solutions. You must know some standard properties of limits for the logarithmic functions to understand how limits rules of logarithmic functions are used in finding limits of logarithmic functions.

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\log_{e}{(\cos{x})}}{\sqrt[\Large 4]{1+x^2}-1}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, e}{\normalsize \dfrac{\log_{e}{x}-1}{x-e}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x^x-1}{x\ln{x}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\log_{\displaystyle e}{\big(\cos{(\sin{x})}\big)}}{x^2}}$