Who has not be amazed to learn that the function y = ex, like a phoenix rising again from its own ashes, is its own derivative?

– Francois le Lionnais

An exponential is anything that grows at a rate proportional to its size. An exponential function is a function of the form y = a^x, where a is an arbitrary constant greater than zero and x is variable between -∞ and +∞.

 

An exponential function has the property:

 

    a^m . aⁿ = a^{m + n}

 

If a = e we have the important exponential function y = e^x, where e is about 2.712... The inverse of this function is y = ln x. The exponential function to base a can be written as f (x) = a^x.

 

It was proved by Leibniz that exponential functions are transcendental.