Applications of Exponentials and Logarithms

An exponential model is simply a formula for anything that grows or shrinks by the same percentage every hour, year or day. The compound interest formula looks crowded, but it's just $P$ (what you start with) times $1 + \frac rn$ (the growth for one period: the yearly rate $r$ split into $n$ pieces), multiplied once per period, $nt$ times. The growth and decay formula is the same idea with $Q_0$ (the starting amount) and $k$ (how fast it changes, negative for decay). These are the formulas behind bank accounts, carbon dating and medicine doses. Nothing here is new; it's the last four lessons put to work.

Continuous compounding. $Pe^{rt}$ is what $P\left(1 + \frac rn\right)^{nt}$ approaches as you compound more and more often. It's the same limit that defined $e$ in the first lesson of this unit, just with a rate $r$ in it.

Doubling time and half-life. "How long until it doubles?" means solve $e^{kt} = 2$ for $t$, and you get $t = \frac{\ln 2}{|k|}$. Half-life is the same with $\frac12$. Notice the starting amount isn't in the answer: an exponential takes the same time to double whether you have $10 or $10 million, because each step multiplies by the same factor. That's also why a substance with half-life $T$ keeps a fraction $\left(\frac12\right)^{t/T}$ after time $t$: $\frac tT$ just counts how many half-lives have gone by.

Finding $k$ from data. If a quantity grows from $Q_0$ to $Q_1$ in time $t_1$, then $Q_1 = Q_0e^{kt_1}$. Divide and take a log, and $k = \frac{1}{t_1}\ln\frac{Q_1}{Q_0}$.

Banks, epidemiologists, pharmacologists and archaeologists (carbon dating) use these exact formulas every day.