Logarithms and Modelling

Many phenomena in nature seem to follow the law that an amount A varies with time according to the formula

A=A₀ e^kt where A₀ is the original amount (the amount at time t=0) and k is a non-zero constant.

If k is positive, then the amount A gets larger or grows with time, and the amount is said to be experiencing exponential growth.
If k is negative, then the amount A gets smaller or diminishes with time, and the amount is said to be experiencing exponential decay.

When solving exponential growth/decay problems, two situations often occur:

• Time is given; you are to find the amount at that given time. This usually just involves evaluating the amount function.

• The amount is given; you are to determine at what time does this amount occur. This usually involves solving an exponential equation, which means logarithms will be needed.

For a straightforward application of these ideas, see the exponential growth problem.

Of course, word problems involving exponential growth and decay may be set up in different ways. Sometimes you will have to determine A₀ or k or both, from the information given in the problem, before you proceed as indicated above. See the radioactive decay example.

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