Log and exponential questions in Additional Math cover solving equations, changing base, using the laws of logarithms, and sketching. The reports show that the method is usually known. The marks go missing in the handling of the laws, the solutions to reject, and the form of the final answer. 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
- Removing the logarithm. Writing x = 4³ instead of x = 3⁴ and getting 81, or giving x = 3 instead of x = 4.
- Incorrect laws. Stating that log_a (5p) = log_a p + log_a 5 − log_a 4 (or similar wrong versions of the laws), which sometimes leads to a fortuitous "correct" answer that earns no credit. Also writing 2 log₅ 2x as log₅ 2x² instead of log₅ 4x², and not using the fact that 1 can be written as log₅ 5.
- Change of base. Incorrect changes of base, which meant little progress could be made, and incorrect manipulation of ln 9 and ln 3 (the reports mention ln₃ 9 and ln₉ 3).
- Squares and logarithms. Writing lg x² = 4 where (lg x)² = 4 is meant. Reports say candidates misinterpreted the square as the square of the argument instead of the square of the logarithm. Replacing lg x with a dummy variable prevents this.
- Solutions to reject and keep. Forgetting to discard x = 1 when it is not valid, and at the other extreme, incorrectly discarding a valid solution such as log₅ x = −1 (x = 1/5). The reports ask candidates to check the validity of every answer in a log question.
- Missing solutions. Substituting y = 3^x from the start led to an equation where many gave only one solution, omitting x = 0. In another question, students gave only x = √5 from the positive root, or did not deal correctly with the negative index of 1/2.
- Algebra in the equation. Sign errors (the most common cause of mistakes in several questions), omitting to multiply a term by 4 when clearing fractions, bracket errors, and writing (2x + 1) ÷ 2.5 instead of 2x + 1 − 2.5. A few could not rearrange y² / (4y − 9) = 3 because of sign slips or omitting to multiply −9 by 3.
- Unhelpful substitutions. Setting u = e³ early in a question, which led to random errors.
- Not finishing. Solving the quadratic for u and not going on to find t, as required.
- Form of the answer. A single exact answer was expected in one question. For non-exact answers, an answer of −0.92 did not gain the accuracy mark unless the more accurate form had been seen. In one question the answer was required as a single logarithm.
- Sketches. Omitting the intercepts with the axes, although many gave the correct equation of the asymptote.
- Early errors. One report says candidates needed great care in the early stages, because no recovery was permitted after an uncorrected error in method.
What examiners advise
- Show every step. Write correct statements in terms of logarithms before you move to an equation in y, as the rubric on the front of the paper says that all necessary working should be shown.
- Use a dummy variable for a repeated logarithm, then solve the quadratic, then convert back.
- Check validity: the argument of a logarithm must be positive.
- Give exact answers when they are asked for and 3 significant figures otherwise.
- Set out each step clearly when you rearrange an equation after changing the base.
Habits that help
- Write "log_a b = c means a^c = b" at the start, then apply it to the equation.
- When you take both sides of an equation, put brackets around every expression.
- Check each solution by substituting it into the original equation, using a calculator if allowed.
Questions students ask
How do I remove a logarithm correctly?
log_a x = b means x = aᵇ. Reports record x = 4³ written instead of x = 3⁴, giving 81. Write correct statements in terms of logarithms before moving to an equation in y.
What is the difference between lg x² and (lg x)²?
lg x² is the logarithm of x squared, and (lg x)² is the square of the logarithm. Candidates misinterpreted the square as the square of the argument instead of the square of the logarithm. Use a dummy variable for a repeated logarithm.
Why must I check validity of solutions?
The argument of a logarithm must be positive. Students forgot to discard an invalid x = 1, and also wrongly discarded a valid solution such as log₅ x = −1, which gives x = 1/5.
What goes wrong when I substitute y = 3ˣ?
Many gave only one solution and omitted x = 0. After solving the quadratic, go on to find the original variable. Setting u = e³ early led to random errors.
How should I give the answer?
In exact form when asked, otherwise to 3 significant figures. An answer of −0.92 did not gain the accuracy mark unless the more accurate form had been seen.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.