About this chapter
Exponentials and logarithms rewards students who know the laws of logarithms exactly and punishes those who almost know them. The examiner reports for June 2023 and June 2024 describe the same slips year after year: splitting a log across a bracket, combining logs before using the power law, and keeping or rejecting solutions without checking them.
This chapter covers exponential graphs and their asymptotes, converting between index and log form, the laws of logarithms, solving equations with logs, change of base, hidden quadratics in powers of 2, and where two exponential curves meet. The last questions turn log equations into quadratics and ask you to show which answers are valid.
The ten questions
- Q1 (4 marks): the value of log to base 3 of 81, then combine three logs to base a into a single log.
- Q2 (5 marks): solve 5 to the power x = 12, then an equation with powers of 3 and 7, both by taking logs.
- Q3 (6 marks): a log equation with three brackets leads to an answer of the form a + √b, and any other root must be rejected with a reason.
- Q4 (5 marks): sketch y = 4 minus 2 to the power x with its asymptote, then an exact meeting point with y = 1.
- Q5 (5 marks): show that a log to base 4 is half the log to base 2, then solve an equation using both.
- Q6 (8 marks): a hidden quadratic with y = 2 to the power x, then when the expression is negative.
- Q7 (8 marks): where the curves for powers of 5 and powers of 2 cross, and the range where one is above the other.
- Q8 (8 marks): a log in terms of p and q, then simultaneous log equations.
- Q9 (9 marks): an equation mixing a log to base a with a log to base y becomes a quadratic, then a condition fixes a.
- Q10 (10 marks): a log equation gives x² − 8x + 8 = 0 with surd answers, then a substitution solves a second equation.
Key skills tested
Exponential graphs. A graph of a to the power x passes through (0, 1) with asymptote y = 0. Transformations move the asymptote, so 4 minus 2 to the power x has asymptote y = 4.
Index and log form. a to the power x equals b exactly when x is the log of b to base a. For example, the log of 8 to base 2 is 3, because 2³ = 8.
Laws of logarithms. log xy = log x + log y, log(x/y) = log x − log y, and the log of x to the power k is k log x. But log(x+y) is not log x + log y.
Log equations. Combine into a single log on each side, then remove the logs. Check each answer, because you can only take the log of a positive number.
Solving exponential equations. Take logs of both sides and divide. For example, 3 to the power x = 20 gives x = log 20 ÷ log 3 = 2.73.
Hidden quadratics. 4 to the power x is the square of 2 to the power x, so let y = 2 to the power x, remembering that y must be positive.
Change of base. A log to base a equals the same log to base c divided by the log of a to base c. So the log of x to base 4 is half its log to base 2.
Intersections of exponentials. Set the two expressions equal, collect the powers into a single power of a fraction, then take logs.
Substitution in log equations. "Hence" may mean replacing x by 2y, or similar, in an equation you have already solved.
Worked example
Question 2 from this chapter. Find, to 3 significant figures, the value of x for which (a) 5 to the power x = 12, and (b) 3 to the power (2x − 1) = 7 to the power x.
(a) Take logs of both sides: x log 5 = log 12, so x = log 12 ÷ log 5 = 1.54.
(b) Take logs of both sides and bring the powers down: (2x − 1) log 3 = x log 7. Expand and collect the x terms: 2x log 3 − x log 7 = log 3, so x(2 log 3 − log 7) = log 3.
Since 2 log 3 = log 9, x = log 3 ÷ (log 9 − log 7) = 4.37.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2023, Q5: combining logarithms into one, forming a quadratic and rejecting an invalid root, as in Q3.
- June 2024, Q3: a log equation leading to a quadratic, where checking which root is valid decided the final mark, as in Q3 and Q10.
- June 2024, Q6(a): sketching an exponential graph with a horizontal asymptote that is not the x-axis, as in Q4.
- June 2024, Q10(c): two exponential models set equal and solved with logs, as in Q7.
Where marks are lost
- Splitting a log across a bracket. In June 2024 many wrote log(x − 2) as log x − log 2, and never reached the quadratic.
- Combining before the power law. A term like 2 log(x−2) must become log of (x−2)² before it is combined with anything else.
- Rejecting the wrong root. In June 2024 the valid answer was negative, and many students threw it away on the assumption that x cannot be negative. In June 2023 others kept a root that was invalid. Substitute each root back into the original logs.
- The asymptote's label. In June 2024 some labelled the y-intercept with the height of the asymptote, or drew the asymptote without its equation.
Common questions
How do I know whether to reject a solution?
Put it back into every log in the original equation. It is invalid only if one of them would need the log of zero or a negative number.
Can I use my calculator's equation solver?
Not on its own. In June 2023, answers that came straight from a solver with no log working earned little credit. Show the laws of logs and the quadratic first.
What is the quickest way to handle different bases?
Use change of base to write every log in the same base before combining them.
Where this chapter leads
- Pure 2 Chapter 5: Sequences and Series: logarithms find the number of terms in geometric problems.
- Pure 3 Chapter 5: Exponentials and Logarithms: e to the power x, natural logarithms and modelling.