The existence of e is implicit in John Napier\'s 1614 work on logarithms, and natural logarithms. The symbol e for the base of natural logarithms was first used by the Swiss mathematician Leonhard Euler in a 1727 or 1728 manuscript called (Meditation on experiments made recently on the firing of cannon) Euler also used the symbol in a letter written in 1731, and e made it into print in 1736, in Euler\'s Mechanica. There were few assumptions about what the letter e stand for some says that e was meant to stand for "exponential"; others have pointed out that Euler could have been working his way through the alphabet, and the letters a, b, c, and d already had common mathematical uses. What seems highly unlikely is that Euler was thinking of his own name, even though e is sometimes called Euler\'s number.

Euler\'s interest in e stemmed from the attempt to calculate the amount that would result from continually compounded interest on a sum of money. The limit for compounding interest is, in fact, expressed by the constant e.
"e" is a numerical constant that is equal to 2.71828 The value of "e" is found in many mathematical formulas such as those describing a nonlinear increase or decrease such as growth or decay (including compound interest) "e" also shows up in some problems of probability, some counting problems and so many other uses in mathematical problems Because it occurs naturally with some frequency in the world, "e" is used as the base of natural logarithms. e is usually defined by the following equation:

An effective way to calculate the value of e to use the following infinite sum of factorials. Factorials are just products of numbers indicated by an exclamation mark. For instance, "four factorial" is written as "4!" and means 1󫎿4 = 24.
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

The sum of the values is 2.7182818284590452353602875 which is "e."


ex as a function:
The derivative of ex
d
dx ex = ex
"The derivative of ex with respect to x
is equal to ex."

Therefore on taking the derivative of both sides with respect to x, and applying the chain rule to ln y:
= 1.

y\' = y.
That is,
= ex.

(Spector, Lawrence.( 2015 ) the math page)
It implies the meaning of exponential growth. For we say that a quantity grows "exponentially" when it grows at a rate that is proportional to its size. The bigger it is at any given time, the faster it\'s growing at that time
Graph y = ex


Applications on the function of ex :
The number e does have physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduces. Exponential Decay as it similar with population growth. The best thing about exponential functions is that they are so useful in real world situations. Exponential functions are used to model populations, carbon date artifacts, help coroners determine time of death, compute investments, as well as many other applications.
Example 1: in the case when the ratio is 1 (simple interest = 100% of original amount):
Question: If you would earn 100% interest (i.e., your money would double) under simple interest, how much money would you end up with under compound interest?
Answer: You would have e times your original amount.
Example 2: The population of a city is P = 250,342e0.012t where t = 0 represents the population in the year 2000.
Find the population of the city in the year 2010.
To find the population in the year 2010, we need to let t = 10 in our given equation.
P = 250,342e0.012 (10) = 250,342e0.12 = 282,259.82
Since we are dealing with the population of a city, we normally round to a whole number, in this case 282,260 people.
This gives us the following physical meaning for the number e: The number e is the factor by which a bank account earning continually compounding interest or a reproducing population whose offspring are themselves capable of reproduction, or any similar quantity that grows at a rate proportional to its current value or the decay at a rate of proportional to its current value (will increase or decrease), if, without the compounding (or