Limits of logarithmic functions

There are two fundamental properties of limits to find the limits of logarithmic functions and these standard results are used as formulas in calculus for dealing the functions in which logarithmic functions are involved.

$(1) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}}$ $\,=\,$ $1$

The limit of quotient of natural logarithm of $1+x$ by $x$ is equal to one.

$(2) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{b}{(1+x)}}{x}}$ $\,=\,$ $\dfrac{1}{\log_{e}{b}}$

The limit of ratio of logarithm of $1+x$ to a base to $x$ is equal to reciprocal of natural logarithm of base.

Use

Let’s learn how to use these two limits formulas for the logarithmic functions in finding the limits of functions in which logarithmic functions are involved.