Definition of Exponential Function : The exponential function with base a is denoted by f( x ) = b^x, where b > 0, b ≠ 1, and x is any real number.

An exponential function is a function of the form y = b^x.y,

where b is a positive number not equal to 1, and x is any real number.

Thus, exponential functions have a constant base; the variable is in the exponent.

The number b is called the base of the exponential function.

The most important exponential function is when the base is the irrational number e. (Note: e≈2.71828) \ \ In this case, the function is also written as exp(x), and is called the natural exponential function.

The exponential function is f(x) = (1/e)^x → f(x) = (e^{- 1})^x → f(x) = (e)^{- x}.

Here base is e and its value is 2.718281828 . . . . > 0, and this number is called natural base, and x is any real number.

The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is exactly 1.

The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series.