Exponential Growth and Decay Models

Definition An exponential model describes a quantity $Q$ at time $t$ by

where $a > 0$, $b > 0$, $b \ne 1$ and $t$ is measured in a fixed unit of time. It is a growth model when $b > 1$ and a decay model when $0 < b < 1$. Over every time step of length $h$ the quantity is multiplied by the same factor,

Definition The initial value of the model $Q(t) = a\,b^t$ is $Q(0) = a$, the amount at the moment chosen as $t = 0$. Moving that moment changes $a$ but not $b$. If the quantity is known at two times $t_1 < t_2$, dividing one value by the other gives

which determines $b$, and then $a = Q(t_1)\,b^{-t_1}$.

Definition Suppose a quantity changes by the same percentage in each unit of time, with the rate $r$ written as a decimal, $r > -1$ and $r \ne 0$. Then

The number $b = 1 + r$ is the growth factor per unit of time. A positive $r$ means growth and a negative $r$ means decay, so a loss of 15 percent per year has $r = -0.15$ and $b = 0.85$. In the other direction, a growth factor $b$ corresponds to the percent rate $r = b - 1$.

Definition A discrete rate $r$ applies the change once per unit of time, which gives $Q(t) = a(1 + r)^t$. A continuous rate $k \ne 0$ splits the change into ever smaller steps. Applying the rate $k/n$ a total of $n$ times per unit of time gives the factor $(1 + k/n)^n$, which tends to $e^k$ as $n \to \infty$ for every real $k$. For $k > 0$ this is the continuous compounding limit from the lesson on the natural base e, and the same limit holds for $k < 0$. So a continuous rate gives the model

with growth when $k > 0$ and decay when $k < 0$. The two describe the same function exactly when $e^k = 1 + r$, that is

In calculus $k$ turns out to be the relative rate of change, the rate of change of $Q$ divided by $Q$, as in the lesson on exponential growth and decay.

Definition For a decay model $Q(t) = a\,b^t$ with $0 < b < 1$, the half-life is the time $T > 0$ with $Q(t + T) = \frac{1}{2}Q(t)$ for every $t$. It is the solution of $b^T = \frac{1}{2}$, namely

and for $Q(t) = a\,e^{kt}$ with $k < 0$ it is $T = \ln 2 / (-k)$. The half-life depends neither on $a$ nor on $t$, and the model can be written as

Definition For a growth model $Q(t) = a\,b^t$ with $b > 1$, the doubling time is the time $D > 0$ with $Q(t + D) = 2Q(t)$ for every $t$, namely

and for $Q(t) = a\,e^{kt}$ with $k > 0$ it is $D = \ln 2 / k$. The model can be written as $Q(t) = a \cdot 2^{t/D}$.

Definition A principal $P > 0$ at a nominal annual rate $r > 0$, compounded $n$ times per year with $n$ a positive integer, grows after $t$ years to

and to $A(t) = Pe^{rt}$ under continuous compounding, which is the model $a\,e^{kt}$ with $a = P$ and continuous rate $k = r$. These are growth models with annual growth factors $\left(1 + \frac{r}{n}\right)^n$ and $e^r$. Interest compounded $n$ times a year is credited only at the end of each period, so the formula gives the actual balance only when $nt$ is a whole number of periods.

Definition A logarithmic scale places a positive quantity $x$ at the position $\log x$, so each unit step along the scale multiplies the quantity by $10$. A semi-log graph has a logarithmic vertical axis. For an exponential model

which is a linear function of $t$. So on a semi-log graph an exponential model is a straight line with slope $\log b$ and vertical intercept $\log a$. The pH scale, sound levels in decibels and earthquake magnitudes are logarithmic scales too. Each uses a constant multiple of a logarithm, such as $-\log x$ for pH or $10 \log \frac{I}{I_0}$ for decibels, so on each of them a fixed step stands for a fixed ratio.

Definition An exponential model assumes that the quantity changes by the same percentage in every time step of equal length, for as long as the model is used. The growth factor stays fixed, nothing limits the size of the quantity, and in a discrete model the change happens once per period. A model is fitted to data from a limited range of times, and its predictions outside that range, called extrapolations, are only as good as these assumptions.