Growth by a fixed percent
When a quantity grows by the same percent each period, it's multiplied by the same factor each period. That's exponential growth.
Compound interest
If interest is added \(n\) times a year at annual rate \(r\), each period multiplies the balance by \(1 + \frac{r}{n}\):
How long does it take? Use logarithms
When the unknown is time, it's in the exponent: take a logarithm.
Half-life
A radioactive substance always takes the same time, its half-life \(h\), to lose half its mass:
\[A = A_0\left(\frac{1}{2}\right)^{t/h}\]
Practice
Quick check
Which formula gives the balance with continuous compounding?
Continuous compounding uses the natural base \(e\).
Quick check
A substance has a half-life of \(5\) years. What fraction remains after \(20\) years?
\(20\) years is \(4\) half-lives: \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\).