It is difficult to say whether it is useful to bring exponential growth and decay into an explanation of decibels. To explain decibels adequately, it is necessary to discuss logarithms and logarithmic scaling. The only good way to make sense of logarithms is by first discussing the exponential function, since the logarithm function only becomes intuitive when understood as the inverse of the exponential function. The question thus becomes whether, in order to adequately explain the exponential function (not y = e^x, but rather y = 2^x or y = 10^x), it is desirable to discuss exponential growth and decay. My sense is that in order to demonstrate the behavior of y = 10^x (or y = 2^x), where the independent variable 'x' is the exponent to which some constant is raised, it is probably not necessary to discuss exponential growth and decay. Nevertheless, for anyone who deems this useful and is looking for some good examples:
Exponential growth:
1. Rabbits, mice and cats increasing in population. Most everyone is familiar with how unchecked population growth leads to a doubling in population at fixed time intervals, i.e., the population doubles every three months, or every six weeks, etc.
2. Compound interest. Most everyone has an intuitive understanding of the benefit of compounded interest over time, when the rate is fixed.
3. Many people are familiar with a story about how some king was duped by a clever man who asked for his reward (for something he had done for the king) to be only a single kernel of wheat (or rice) for the first square on a chess board, then two grains the next day for the next square, then four grains the 3rd day, then eight grains the 4th day, etc., doubling the number of grains each day and continuing through 64 days. This is an excellent way to illustrate exponential growth. Note, though, when this example is used, you'll likely not be able to keep from explaining the shortcut for calculating the sum (2^65 -1). This shortcut for calculating the sum isn't an essential aspect of exponential growth, and if you discuss it, it will obfuscate or diminish the essential facts about exponential growth.
4. Then they tell two friends, then they tell two friends, etc.
Exponential decay:
1. Zeno's paradox (the one everyone knows, among the several that were preserved by Plato). If individual steps are taken at constant time intervals, and each step is only half as great as the preceding steps, then distance as a function of time will exhibit exponential decay.
2. Half-life of anything that decays exponentially, but this is best understand from the standpoint of elimination of a drug from the body, i.e., at fixed time increments, the amount remaining in the blood is half the amount that remained at the end of the prior time increment. This is easy to comprehend and is an excellent way to illustrate exponential decay.
Exponential growth: