Limit of logarithmic function

A logarithm function is an exponent of exponential function. For example,
if \( {a^x}=y\), then \(x={\log_a}y\).
In this definition
Log is the exponent, (or, exponent= Log)
if
\(3^2=9\) then \(2 = \log_3 9\)

In general, a function of the form \(f (x) = \log_e x\) called logarithmic function.
where

• Domain of f (x) = \( (0, \infty )\)
• Range of f (x) =\( (-\infty , \infty ) \)

Properties of logarithmic function

1. Product property: \( \log a (x.y) = \log_ax + \log a_y \)
2. Quotient property: \( \log_ a (x/y) = \log_ax - \log_ay \)
3. Power property: \( \log _ ax^nn = n \log _ax \)
4. \( \log_ a a = 1, \log_ a 1 = 0 \)
5. \( \log_ a m = \log_ a b \times \log_ b m \)

Graph of exponential and logarithem function

Log is the reflection of exponential function about y=x line, which is shown in a graph given below

Theorems on Limit of logarithmic function

1. \(\displaystyle \lim_{x\to 0}\frac{\log_e(1+x)}{x}=1 \)
2. \(\displaystyle \lim_{x\to 0}\frac{\log_e(1-x)}{-x}=1 \)
3. \(\displaystyle \lim_{x\to 0}\frac{a^x-1}{x}=\log a\)
   We know that
   \( \frac{d}{dx} a^x= \frac{d}{dx} e^{\log (a^x)} \)
   or \( \frac{d}{dx} a^x= \frac{d}{dx} e^{x \log a} \)
   or \( \frac{d}{dx} a^x= \log a .e^{x \log a} \)
   or \( \frac{d}{dx} a^x= \log a .e^{\log a^x} \)
   or \( \frac{d}{dx} a^x= \log a. a^x \)
   Thus, the limit is
   \(\displaystyle \lim_{x\to 0}\frac{a^x-1}{x}\)
   or \(\displaystyle \lim_{x\to 0}\frac{ \frac{d}{dx}(a^x-1)}{\frac{d}{dx} (x)} \)
   or \(\displaystyle \lim_{x\to 0}\frac{ log a a^x}{1} \)
   or \(\log a .a^0 \)
   or \(\log a \)
   This completes the proof