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3.13 · Paper 2C
Growth and decay, the number e, and using logarithms to solve equations like $a^x = b$.
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Exponential growth means multiplying by the same factor in equal steps: $y = ab^x$, where $a$ is the starting value and $b$ is the multiplier. If $b > 1$ it is growth; if $0 < b < 1$ it is decay. A 3% rise each year has multiplier 1.03; a 12% fall has multiplier 0.88.
The number $e \approx 2.718$ is special: the function $y = e^x$ has a gradient equal to its height at every point. Models of continuous growth or decay are written $N = Ae^{kt}$ — use the $e^x$ button on your calculator.
$\ln x$ is the inverse of $e^x$: $\ln(e^x) = x$ and $e^{\ln x} = x$. To solve $a^x = b$, take logarithms: $x = \dfrac{\ln b}{\ln a}$. This finds, for example, how long money takes to double at a given interest rate.
A town's population is 8,000 and grows by 3% each year. Write a formula for the population $P$ after $t$ years and find $P$ after 10 years.
Multiplier = 1.03, so $P = 8000 \times 1.03^t$.
After 10 years: $P = 8000 \times 1.03^{10} = 8000 \times 1.3439 = $ 10 751 (nearest whole number).
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