Hints and tips - 12 minute read
Steven Walker, Maths Subject Advisor
Logarithms were originally used to multiply very large numbers. They have since become a useful technique to solve equations involving exponential functions, or calculus problems with complex functions. In this blog I look at some of the common errors and misconceptions seen by examiners and provide ideas for supporting students’ understanding.
Scottish mathematician John Napier is credited with developing logarithms to solve complex arithmetic problems in the fields of engineering, navigation and astronomy (he’s also remembered for Napier’s bones and the introduction of the decimal point).
Whilst calculators have now made some of these original arithmetical applications of base 10 logarithms redundant, it is still useful to develop an understand of the underlying principles about converting into logs, performing calculations and then converting back to an answer using antilogs.
A short investigation using four figure logarithm tables (read the notes from Abelard about the use of tables) may help students see beyond the button on the calculator. You may also want to talk about the slide rule – see the slides and video demonstration created by Design News.
Students are generally introduced to logarithms through their link to indices. Any misconceptions that students may have with the rules of indices, especially x0 = 1 and x- 1 = 1/x, will make it difficult to manipulate expressions into logarithm form.
A good refresher may be the Higher tier Check In tests for OCR GCSE (9–1) Mathematics – J560 topic 3 – Indices and surds.
3.01 Powers and roots
3.02 Standard form
3.03 Exact calculations
Khan Academy has published a set of introductory notes to logarithms that may be useful for students looking for support and practice.
Candidates often struggle with identifying the initial algebraic manipulation needed before taking logarithms.
Question 6 Core 2 – 4752 Jan 2012 Mathematics MEI – 3895/7895.
Examiners noted that a common mistake was to immediately take logarithms to give Log y = Log (a + xb), rather than to correctly rearrange into the form y – a = xb before taking logarithms.
Question 8 Core 2 – 4722 Jun 2015 Mathematics – 3890/7890 (edited).
This type of problem could be solved efficiently using either base 2 or base 10 logarithms. Common errors were students not doing the same to both sides of the equation, using different bases on each side, or only taking logarithms on one side.
See Logarithm lattice, on Underground Maths, for a short puzzle investigating the magnitude of terms using different base.
Modelling problems regularly involve geometric progressions.
Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6g of the chemical and in the second experiment she uses 7.8g of the chemical.
It is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that N, the greatest number of experiments possible, satisfies the inequality 1.3N ⩽ 91 and use logarithms to calculate the value of N.
Question 6 Core 2 – 4722 Jun 2013 Mathematics – 3890/7890 (edited)
In this question students often made errors in their algebraic manipulation before taking logarithms, as well as errors with inequality symbols.
This wording directed candidates towards the use of logarithms, but this may not be explicit in the current qualifications where decision making may form part of the problem solving. Students may be tempted to use numerical methods to find a solution through trial and improvement, but this should be discouraged where an efficient analytical technique such as logarithms is available.
Candidates must also take into consideration the context of the question. In this case, the result N = 17.19 would not score full marks since the greatest number of experiments that can be performed is the rounded down value of N = 17.
Logarithms are used in a variety of subjects to simplify functions to simple linear relationships, for example the pH scale in chemistry and the Moment Magnitude scale in geology.
There are two common forms of equation that logarithms can reduce to linear form.
Students need to be aware of these two types and show clear working to avoid algebraic mistakes that could lead to an incorrect relationship.
The data may be represented by an exponential model of growth. Using t as the number of years after 1960, a suitable model is p = a × 10kt.
(a) Derive an equation for log p in terms of a, k and t.
(b) Plot log p against t.
(c) Use your graph to estimate a and k.
(d) Hence estimate the population in 1960.
(e) According to the model, when will the population reach 200 million?
Question 12 Core 2 – 4752 Jan 2013 Mathematics MEI – 3895/7895 (edited).
Candidates are free to make their own choice of base, but in this case base 10 is the most efficient.
This type of question is looking for an estimated line of best fit, but examiners note some candidates do simply connect the points or attempt a freehand curve.
Reducing complex relationships to a linear graph means that the gradient and y-intercept can be used with y = mx + c for to find the constants for the proposed model. While students often use the graph correctly, they can allocate the obtained values to the incorrect constants and so care should be taken.
Questions of this type may also involve some evaluation of the model, where extrapolation may not be valid.
The use of natural logarithm (base e) is common where complex reciprocal functions are integrated. The modulus function is used here rather than brackets, because the logarithm of a negative value is undefined. This means that the issue of modulus rather than brackets is only critical if the nature of the domain is unclear. It is always safer to use the modulus, especially when working with indefinite integrals.
See integration that leads to logarithm functions, from Mathcentre, for some examples and worksheet and also Section 1-4: Partial fractions for interactive examples on Paul’s online notes.
The questions used in this article came from legacy A Level papers. These can be found in the ‘assessment’ resources area for Mathematics A – H240 and Mathematics B (MEI) – H640, within the ‘withdrawn qualification materials’ section. Editable versions can be found on ExamBuilder, our free question building platform.
Delivery guides and ‘section check in tests’ can be found in the ‘planning and teaching’ area for the qualifications at the links below.
For some examples of cross-curricular background reading (and those students thinking ahead to HE courses)
There are so many great online resources available, so join the conversation by sharing your ideas and links to all your favourites in the comment box below.
If you have any questions, email us at firstname.lastname@example.org, call us on 01223 553998 or tweet us @OCR_Maths. You can also sign up for email updates to receive information about resources and support.
Steven joined OCR in 2014 during the major qualification reform period and now primarily focuses on supporting the Level 3 maths qualifications. Steven originally studied engineering before completing a PGCE in secondary mathematics. He began his teaching career with VSO in Malawi and has taught maths in both the UK and overseas.