CAT 2025 Lesson : Logarithm - Exponential Function

2. Exponential Function

For any real number $x$, $e^{x} = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}$ $= \dfrac{x^{0}}{0!} + \dfrac{x^{1}}{1!} + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + ...$

$e^{x} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + ...$

While complex questions pertaining to the exponential function are not expected, there could be questions pertaining to basic properties. These listed below are derived from the above equation.

$1$) $e^{0} = 1$
$2$) $e = 1 + 1 + \dfrac{1}{2!} + \dfrac{1}{3!} + ... \thicksim 2.71828$
$3$) $e$ is to be treated as a constant in arithmetic operations.
$e^{2} \times e^{3} = e^{5}$
$(e^{5})^{10} = e^{50}$
$\dfrac{e^{2}}{e^{3}} = e^{-1}$
$4$) Given $e$ is a positive constant, where $x$ is a real number (positive or negative) $e^{x}$ is always positive, i.e. $e^{x} > 0$

The graph of the exponential function and logarithmic functions are as below.