## Exponential Functions

Exponential growth means growth without limit. The rate of growth of a quantity is directly proportional to the quantity itself and this leads to equations of the form where and are constants, represents the quantity and represents the time. In the long term of course, nothing ever grows without limit, a lesson bankers are learning and exponential growth functions can only apply over certain ranges. If a question ever asks, why is this wrong, and you have arrived at an exponential function, the answer is probably because in the long term exponential growth functions are impossible.

Exponential decay functions ARE perfectly possible however. The best example is probably the exponential decay curve. The quantity of a radioactive material decays smoothly to zero, and zero is a very plausible quantity to have. If we know the exponential function we can find the quantity present at any time by substituting the value of into the expression for the quantity. For example, if we can find when by calculating to 4 significant figures. If instead we know the quantity present we can find the value of t by using If then  