About this chapter
Pure 3 exponentials and logarithms moves from the logs of Pure 2 to the natural base e and its inverse, ln. Questions expect exact answers written in terms of e or ln, and a large share are models: a cooling drink, a bird population, subscribers to a service, each with a rate of change and a long-term limit.
The chapter covers the graphs of e to the power x and ln x, their inverse relationship, the laws of logarithms, exact solutions, hidden quadratics in e to the power x, exponential models and their rates of change, long-term behaviour, and straight-line log graphs used to find a and n, or a and b, in a model.
The ten questions
- Q1 (5 marks): an exponential equation solved exactly as a + ln b, then a log equation giving x = 3/(e−2).
- Q2 (5 marks): 2 e to the power x plus 3 e to the power −x equals 7, solved as a hidden quadratic.
- Q3 (5 marks): sketch y = 3 − 2 e to the power −x with its asymptote, then its range for x ≥ 0.
- Q4 (5 marks): a straight line of log y against log x gives y in the form a times a power of x.
- Q5 (6 marks): sketch a log-log graph, then show N = a times b to the power t from a line on a base 3 log graph.
- Q6 (10 marks): a cooling coffee model: find A, show k = (1/5) ln 2, a time, and a rate of cooling.
- Q7 (10 marks): a bird population model: the start, show k, the long-term number, and an exact time.
- Q8 (10 marks): subscribers modelled as N = a times b to the power t from a straight log graph, with b interpreted.
- Q9 (10 marks): sketch f(x) = ln(2x − 4) + 1, its inverse with domain and range, and a composite equation.
- Q10 (12 marks): two insect populations: when they are equal via a quadratic, an exact time, and a rate of change.
Key skills tested
Graphs. y = e to the power x has asymptote y = 0 and passes through (0, 1). y = ln x has asymptote x = 0 and passes through (1, 0). Transformations move the asymptotes too.
Inverse functions. ln and e to the power x undo each other, and ln x only exists for positive x.
Laws of logarithms. ln a + ln b = ln ab, ln a − ln b = ln(a/b), and k ln a is the log of a to the power k.
Exact solutions. Take ln of both sides and leave the answer in terms of ln or e. For example, 3 e to the power (2x − 1) = 12 gives 2x − 1 = ln 4.
Hidden quadratics. Multiply through by e to the power x and let u be e to the power x. Reject any u that is zero or negative, since exponentials are always positive.
Exponential models. The initial value comes from t = 0, and a known point finds k exactly, such as k = (1/5) ln 2.
Rates of change. Differentiate: the rate for A e to the power kt is kA e to the power kt. A negative rate means decreasing, so say so in context.
Long-term behaviour. For positive k, e to the power −kt tends to 0, so the model tends to a limit. State limitations in context.
Logarithmic graphs. y = a times x to the power n gives a straight line of log y against log x with gradient n. y = k times b to the power x gives a straight line of log y against x. Use the base on the axis.
Worked example
Question 1 from this chapter. Find the exact solutions of (a) 3 e to the power (2x − 1) = 12, in the form a + ln b, and (b) ln(2x + 3) − ln x = 1.
(a) Divide by 3: e to the power (2x − 1) = 4. Take ln: 2x − 1 = ln 4, so x = ½ + ½ ln 4. Since ½ ln 4 = ln 2, the answer is x = ½ + ln 2.
(b) Combine the logs: ln((2x + 3)/x) = 1, so (2x + 3)/x = e. Then 2x + 3 = ex, so x(e − 2) = 3 and x = 3/(e − 2).
Both answers stay exact. Turning to a calculator for a decimal here would lose the final mark.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q3: sketching a log graph from a straight-line relationship, then a base 3 log graph turned into an equation for N in terms of t, as in Q4, Q5 and Q8.
- June 2024, Q5(a): an equation solved exactly with e and ln, as in Q1.
- June 2024, Q8(a): a model with e set equal to zero and solved with logs under a calculator warning, like the models in Q6, Q7 and Q10.
Where marks are lost
- The wrong base. On the June 2024 log graph, the most common error was using base 10 instead of base 3.
- Values but no equation. Many found a and b and stopped, when the question needed the final equation written out.
- A decimal instead of an exact answer. In June 2024 a disappointing number reached for a calculator rather than simplifying the power to an exact value.
- Taking logs too early. Students who took logs before rearranging the June 2024 model often misapplied the log laws. Rearrange to a single exponential first.
Common questions
What does "exact" mean here?
Leave the answer in terms of ln or e, such as ½ + ln 2 or 3/(e − 2), not a rounded decimal.
Why reject a negative value in a hidden quadratic?
Because e to the power x is always positive, so it can never equal a negative number.
Which log graph gives which model?
If log y against log x is a straight line, the model is a times x to the power n. If log y against x is a straight line, it is k times b to the power x.
Where this chapter leads
- Pure 3 Chapter 6: Differentiation: differentiating exponentials and logs with the chain, product and quotient rules.
- Pure 4 Chapter 6: Integration: differential equations whose solutions are exponential models.