Exponential and Logarithmic Series

The Number e

The sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...¥ is called the exponential number and is denoted by 'e'.

Exponential Series

If x is any complex number then the series is called the exponential series. It can be proved mathematically that this exponential series has a sum and we denote it by e^x.

Exponential Theorem

If a > 0, then prove that

Graph of Exponential Function

We see that as x increases, the value of e^x also increases indefinitely. Also, as x decreases, the value of e^x tends toward zero. The function e^x is one-one.

Exponential and Logarithmic Function

The function e^x (where e is the number approximately 2.718) is called exponential function.
and the functions of the form y= log_bx where b > 0 and b ≠ 1 is known as logarithmic functions.

Some Particular Exponential Series

The following are the Some Particular Exponential Series:

Graph of Logarithmic Series

We see that as x increases from 0 to ¥, the value of log x also increases indefinitely. The function log x is one-one.

Logarithmic Series

If x is a real number such that |x|<1,>