Exponential and logarithmic models

A relation of the form y = a · bᵗ is called an exponential model: a is the initial value (y = a at t = 0) and b is the multiplier. If b > 1 the quantity grows (exponential growth); if 0 < b < 1 it decays. For an increase of p % each year the multiplier is b = 1 + p/100, for a decrease of p % it is b = 1 − p/100; after x years y = a · bˣ. If the mass of a substance halves every T (the half-life), m(t) = m₀ · (1/2)^(t/T). A logarithmic model serves the inverse questions: the time needed for the quantity to change k times is x = log_b k. The magnitude of an earthquake is R = lg I (I is the intensity): when the magnitude rises by 1 the intensity grows 10 times, and when it rises by 2 it grows 100 times. We must remember the range in which a model works: nothing in nature grows without limit.