SignificanceExponential functions are functions of the form f(x) = bx for a fixed base b which could be any positive real number. Exponential functions are characterized by the fact that their rate of growth is proportional to their value. For example, suppose we start with a population of cells such that its growth rate at any time is proportional to its size. The number of cells after t years will then be at (an exponential function) for some a>0.
Standard NotationOne exponential function, f(x)=ex, is distinguished among all exponential functions by the fact that its rate of growth at x is exactly equal to the value ex of the function at x. The number "e" is named after Leonid Euler.
Rules of ExponentiationThe main rules used in manipulating exponential functions are:
Graphs of Exponential Functions
different exponential functions are really the same with a scaling of the x-axis. If the scaling is negative, the x-axis is flipped, which accounts for the different behavior.
Exponential Growth and DecayA function whose rate of change is proportional to its value exhibits exponential growth if the constant of proportionality is positive and exponentional decay if the constant of proportionality is negative. For exponential growth, the function is given by kbx with b > 1, and functions governed by exponential decay are of the same form with b < 1. Populations might exhibit exponential growth in the absence of constraints, while quantities of a radioactive isotope exhibit exponential decay.
Exponential EquationsThese are done on the nex page.