F: Exponentials and logarithms: Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models.

The idea

Exponential models describe quantities changing at a rate proportional to their current size. You use y = A e^{kt} or y = A b^t, where A is the initial value and k (or ln b) is the continuous rate constant. For growth, k > 0; for decay, k < 0. In continuous compound interest, A is the principal, k the nominal annual rate, and t time in years. Radioactive decay uses half-life to find k = ln(0.5)/half-life. Drug concentration often decays exponentially after absorption. Population growth may be modelled with k = ln(growth factor per year). Limitations include finite resources, so refinements may use logistic models or piecewise exponential models. You must interpret parameters in context, solve for unknowns using logarithms, and critique assumptions such as constant rate, unlimited growth, or no external factors.

What earns the marks

• Recognise when a situation is modelled by exponential growth or decay and select an appropriate form such as y = A e^{kt} or y = A b^t.
• Interpret the parameters A and k (or b) in the context of the problem, including initial value, continuous rate, and growth/decay factor.
• Use given data to determine the model, for example by using two data points to form simultaneous equations or by using a half-life to find k.
• Solve problems involving exponential models, including finding times for a quantity to reach a given value, using logarithms where necessary.
• Critique exponential models by identifying limitations such as unrealistic long-term behaviour, and suggest refinements such as a logistic model or a restricted domain.

Common mistakes and how to put them right

• Confusing the continuous rate k with the discrete growth factor b.

  Put it right: b = e^k, so if a quantity doubles each year, b = 2 and k = ln 2 ≈ 0.693.

• Forgetting to convert time units or using inconsistent units.

  Put it right: Ensure t and k use the same time unit, for example if k is per year, t must be in years.

• Assuming exponential growth continues indefinitely without considering limiting factors.

  Put it right: State that real populations have carrying capacity, so a logistic model may be more appropriate for long-term predictions.

Exam tips

• Always define your variables and state the model you are using before substituting values.
• When solving for time, take natural logarithms of both sides and show your working clearly.
• In modelling questions, comment on the validity of the model in context, referring to the assumptions and possible refinements.