Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are typically written in the form f(x) = a * b^x, where a is a constant, b is the base (a positive number), and x is the exponent. These functions grow rapidly, meaning that as x increases, the value of f(x) increases much faster than linear functions. This characteristic makes them useful in various fields, such as finance for calculating compound interest and in biology for modeling population growth. One of the most common exponential functions is the natural exponential function, represented as f(x) = e^x, where e is approximately equal to 2.718. This function is significant in calculus and appears in many