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Exponential function IS JUST another function in mathematics study. To date, most of the time we are dealing with so called power function e.g  f(x) = x2
where x is the base
the number 2 is the power (index)
However, in exponential function we are dealing  f(x) = 2x
where 2 is the base
the number x is the power (index)
source: Wikipedia

The big difference is in that the variable is now the power, rather than the base.

source: Wikipedia
The most commonly encountered exponential-function base is the transcendental number e , which is equal to approximately 2.71828. Thus, the above expression becomes:(see also natural logarithm) 
f (x) = e x

Value of  e
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