Logarithmic graphs and modelling growth and decayEdexcel A-Level Maths: Revision notes

Relationships of the form $y=ax^{n}$ become linear when you take logarithms of both sides: $\log y=n\log x+\log a.$ Plot $\log y$ (vertical) against $\log x$ (horizontal). The points lie on a straight line with gradient $n$ and vertical intercept $\log a$. Read off the gradient and the intercept from the line, then find $a=10^{\text{intercept}}$ (or $e^{\text{intercept}}$ if you used $\ln$). Example: gradient $2.5$ and intercept $0.6$ (using $\log_{10}$) gives $n=2.5$ and $a=10^{0.6}=3.98$, so $y=3.98x^{2.5}$.

For $y=kb^{x}$, take logarithms: $\log y=x\log b+\log k$. Plot $\log y$ against $x$ (not $\log x$): the line has gradient $\log b$ and intercept $\log k$. So $b=10^{\text{gradient}}$ and $k=10^{\text{intercept}}$. Example: gradient $0.30$ and intercept $1.20$ gives $b=10^{0.3}=2.00$ and $k=10^{1.2}=15.8$, so $P=15.8\times2.00^{t}$. If the horizontal axis is $\log x$ the relationship is a power law; if it is $x$ the relationship is exponential.

Exponential growth and decay are modelled by $N=Ae^{kt}$, where $A$ is the initial value (at $t=0$): $k>0$ gives growth and $k<0$ gives decay. Examples include continuous compound interest, radioactive decay, drug concentration decay and population growth. The rate of change is proportional to the current value. For large $t$: growth without bound if $k>0$; if $k<0$, $e^{kt}\to0$ so $N\to0$. Example: $C=40e^{-0.25t}$ starts at $40$ and tends to $0$.

Substitute known values into the model to find constants, then solve using logarithms. If $40e^{-0.25t}=10$, then $e^{-0.25t}=\frac14$, so $-0.25t=\ln\frac14$ and $t=\frac{\ln4}{0.25}=5.55$. For a half-life, set $N=\frac{A}{2}$: $e^{kt}=\frac12$, so $t=\frac{\ln2}{-k}$, which does not depend on $A$. For a doubling time, $t=\frac{\ln2}{k}$. To use a straight-line graph: take $\ln$ of $N=Ae^{kt}$ to get $\ln N=\ln A+kt$, so $\ln N$ against $t$ has gradient $k$ and intercept $\ln A$.

Exponential growth is unbounded, so it fails for large $t$: a population cannot exceed what its environment can support. For example, $P=12e^{0.08t}$ predicts $35.8$ million birds when $t=100$. Comment on validity by comparing the prediction with a realistic limit, and suggest a refinement: a model with an upper limit (growth that slows as resources run short), or different constants for different time periods. Models may also be unreliable for small values, or beyond the range of the data used to fit them (extrapolation).