Exponentials and logarithms

What exponentials and logarithms covers

The mathematics of compounding, and a Year 12 unit of only three lessons. One base grows at a rate equal to its own value, logarithms undo powers and bring an unknown down out of an exponent, and a log-scaled plot turns curved data into a straight line whose gradient and intercept can be measured. Everything here returns in differentiation, integration and the differential equation work.

The main ideas

• The exponential curve for any positive base other than 1, growth and decay alike, with the point (0, 1) and the horizontal asymptote labelled.
• The number e, the gradient property that singles it out, and transformations of an exponential curve with the asymptote moved to match.
• Logarithms as the inverse of an exponential, natural logarithms as the inverse of the one to base e, and the translation between the two written forms.
• The three log laws, applied by name, and equations with the unknown sitting in the exponent.
• Linearising a power law and an exponential with the right choice of log plot, and reading the constants off the gradient and the intercept.
• Growth and decay in context: initial values, long-term behaviour, half-life, and where the model stops describing the situation.

The results it turns on

log_ax = n and aⁿ = x are the same statement

a logarithm is an exponent

log xy = log x + log y, log(x/y) = log x − log y, log x^k = k log x

the three laws, quoted by name in working

the gradient of e^kx is ke^kx

the property that makes e the natural base to model with

P = Ae^kt, with A the value at t = 0

the standard growth and decay model

log y = n log x + log a, and log y = x log b + log k

the two linearisations behind a log-log and a log-linear plot

t = ln 2/k

the halving or doubling time from a rate constant

Where it usually goes wrong

• There is no law for the logarithm of a sum. When an addition turns up inside a log, look for a factorisation rather than a law.
• The intercept of a logged plot is log a or log k, so it has to be un-logged before the constant is quoted.
• The inverse relation applies to a whole side of an equation rather than term by term, so taking ln of each term separately is not a legal step.
• An exponential curve stays above the axis and approaches its asymptote without meeting it, which is what a question about a population that stays above zero is asking you to describe.

Where to start

Exponential functions and e first, because the gradient property is the reason the base matters. Logarithms and their laws next, and give the laws real practice, since every later equation with an unknown exponent runs through them. Log graphs and models last.