Everything about The Exponential Function totally explained
The
exponential function is a
function in
mathematics. The application of this function to a value
x is written as
exp(x). Equivalently, this can be written in the form
ex, where
e is a mathematical constant, the
base of the natural logarithm, which equals approximately 2.718281828, and is also known as
Euler's number.
As a function of the
real variable
x, the
graph of
y=
ex is always positive (above the
x axis) and increasing (viewed left-to-right). It never touches the
x axis, although it gets arbitrarily close to it (thus, the
x axis is a horizontal
asymptote to the graph). Its
inverse function, the
natural logarithm, ln(
x), is defined for all positive
x.
The exponential function is occasionally referred to as the
anti-logarithm. However, this terminology seems to have fallen into disuse in recent times.
Sometimes, especially in the
sciences, the term
exponential function is more generally used for functions of the form
kax, where
a, called the
base, is any positive real number not equal to one. This article will focus initially on the exponential function with base
e, Euler's number.
In general, the
variable x can be any real or
complex number, or even an entirely different kind of mathematical object; see the
formal definition below.
Properties
Most simply, exponential functions multiply at a constant rate. For example the population of a bacterial culture which doubles every 20 minutes can (approximatively, as this isn't really a continuous problem) be expressed as an exponential, as can the value of a car which decreases by 10% per year.
Using the natural logarithm, one can define more general exponential functions. The function
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