Logarithmic and exponential functions in A Level Math Paper 3: common mistakes and examiner advice

Logarithms and exponentials appear in Paper 3 as equations, as straight-line graph questions, as inequalities and as the setting for calculus. The reports show that the same few mistakes with the laws of logarithms cost marks year after year. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Treating the logarithm of a sum as a sum of logarithms. This is the most repeated error in the reports, in many forms: log(a + b) = log a + log b, ln(x − 5) = ln x − ln 5, ln(eˣ + 3) = ln eˣ + ln 3, or splitting ln(1 − e⁻²ˣ) into ln 1 − ln e⁻²ˣ. Logarithms turn products and quotients into sums and differences, but they do nothing to a sum inside the bracket.
• Index slips. Statements such as 8 × 3ˣ = 24ˣ, 4 × 5ˣ = 20ˣ, 3³ = 9, or 4(eˣ − 3) = 4eˣ − 3 were all listed as common. When the bases differ, the powers cannot be combined that way.
• Missing brackets. For example, (3x − 1) ln 4 written as 3x − 1 ln 4. Reports record that many students recovered later, but some did not.
• Not seeing the hidden quadratic. Equations such as 3²ˣ = 3ˣ + 12, or equations containing eˣ and e²ˣ, need a substitution such as y = 3ˣ (or eˣ). Students who did not spot this started with an invalid use of logarithms, such as writing 9 = 3 + 12.
• Keeping an impossible root. After finding the roots of the quadratic, students tried to take the logarithm of a negative number, or did not reject the negative root.
• Not finishing the answer. Students left the answer as a ratio of logarithms instead of completing the solution to the form required, or did not give the answer in the exact form the question asked for.
• Wrong base. Reports mention students who used e where the question said "log" (base 10), and the reverse. Know which base goes with "ln" and which with "log".
• Straight-line graph questions. Students used the values of ln x or ln y as if they were x or y, substituted the coordinates incorrectly into the equation, or compared the equation with y = mx + c incorrectly. Common results were gradients of the wrong sign or missing a factor, such as ±2 ln 3 or ±2x ln 3. Others did not follow the method the question requested, and started from values on the graph in place of the line.
• Inequalities with a modulus. Treating |2x + 1 − 2| < 0.5 as one-sided, or forgetting to combine critical values into one region.
• Accuracy. Questions often ask for 3 significant figures and some students gave 3 decimal places (or the reverse), rounded in the middle, or truncated.

What examiners advise

• Show the step from your quadratic to the solution. Going straight from the quadratic to the answer risks the method mark if there is an error.
• Take logarithms at the start when the unknown is in an index, rather than separating the indices first. One report says the first group were usually successful and the second made more errors.
• Reject values that are not possible, and say so.
• Give the answer in the form requested, whether exact, to a number of significant figures, or as a decimal.
• Use the method the question asks for. If it tells you to use a graph or table, start there.

Habits that help

• Write out the three laws of logarithms and the rule for powers before a question, and check each step against them.
• Check the answer by substituting it back into the original equation.
• For straight-line graphs, write the equation in the form ln y = (something) x + (something) before reading off the gradient and intercept.

Questions students ask

Is ln(a + b) the same as ln a + ln b?

No. Treating the logarithm of a sum as a sum of logarithms is the most repeated error in the reports. Logarithms turn products and quotients into sums and differences, not sums into sums. Examples recorded include ln(x − 5) = ln x − ln 5 and ln(eˣ + 3) = ln eˣ + ln 3.

How do I solve an equation like 3²ˣ = 3ˣ + 12?

Spot the hidden quadratic and substitute y = 3ˣ (or y = eˣ for equations with eˣ and e²ˣ). Students who missed it started with an invalid use of logarithms. After solving, reject any negative root, because you cannot take the logarithm of a negative number.

What is the difference between ln and log in an exam question?

ln is base e and log is base 10. Reports record students using e where the question said "log" and the reverse, so check which one the question uses.

How do I find the gradient and intercept from a straight-line graph question?

Write the equation in the form ln y = (something)x + (something) first, then compare with y = mx + c. Reports record students using ln x or ln y values as if they were x or y, and giving gradients of the wrong sign or missing a factor such as 2 ln 3.

Why do I lose marks when the unknown is in an index?

Often because indices were separated first. One report says candidates who took logarithms at the start were usually successful and those who separated the indices first made more errors. Also write 8 × 3ˣ correctly, since it does not equal 24ˣ.

How should I give the final answer?

In the form asked: exact, to 3 significant figures or as a decimal. Do not leave a ratio of logarithms unfinished, and do not give 3 decimal places when 3 significant figures were asked.