A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Laws of Logarithms — mark scheme explained
The short answer
The laws of logarithms are essential tools in simplifying and solving equations involving logarithmic expressions. These laws allow us to manipulate logarithms in a way that makes them easier to handle. Let's explore each law in detail: 1.
The question
Simplify log 2 (32 × 8)
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
Marks are typically awarded for correctly applying the appropriate law of logarithms, showing intermediate steps, and providing the final simplified answer. Ensure you show all your working to maximize your marks.
What the command words demand
- Simplify
- Simplify the given logarithmic expression using the laws of logarithms.
- Evaluate
- Calculate the value of the given logarithmic expression.
- Express
- Write the given logarithmic expression in a simpler form.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: For questions involving the laws of logarithms, aim to spend about 2-3 minutes per mark. This will give you enough time to carefully apply the laws and check your work.
- Apply the product rule: log 2 (32 × 8) = log 2 32 + log 2 81 mark
- Calculate each logarithm: log 2 32 = 5 and log 2 8 = 31 mark
- Add the results: 5 + 3 = 81 mark
Final answer: 8
Work through every step correctly and you earn all 3 marks.
Another worked example
Simplify log 10 (1000/10)
- Apply the quotient rule: log 10 (1000/10) = log 10 1000 - log 10 101 mark
- Calculate each logarithm: log 10 1000 = 3 and log 10 10 = 11 mark
- Subtract the results: 3 - 1 = 21 mark
Final answer: 2
Work through every step correctly and you earn all 3 marks.
Common mistakes
Forgetting to apply the product rule when simplifying log a (xy)
Why it happens: Students often overlook that the logarithm of a product can be split into the sum of logarithms.
Fix: Always remember to use the product rule: log a (xy) = log a x + log a y
Forgetting to apply the quotient rule when simplifying log a (x/y)
Why it happens: Students sometimes forget that the logarithm of a quotient can be split into the difference of logarithms.
Fix: Always remember to use the quotient rule: log a (x/y) = log a x - log a y
Forgetting to apply the power rule when simplifying log a (x k )
Why it happens: Students often forget that the logarithm of a number raised to a power can be simplified by multiplying the exponent with the logarithm.
Fix: Always remember to use the power rule: log a (x k ) = k log a x
Incorrectly applying the product rule as log a (xy) = log a x × log a y
Why it happens: Students sometimes confuse the product rule with multiplication of logarithms.
Fix: The correct form is log a (xy) = log a x + log a y
Incorrectly applying the quotient rule as log a (x/y) = log a x / log a y
Why it happens: Students sometimes confuse the quotient rule with division of logarithms.
Fix: The correct form is log a (x/y) = log a x - log a y
Forgetting to change the sign when applying the power rule with a negative exponent
Why it happens: Students sometimes forget that a negative exponent results in a negative logarithm.
Fix: Always remember to use the correct form: log a (x -1 ) = -log a x
Using the wrong base for logarithms in calculations
Why it happens: Students sometimes use a different base than specified in the problem.
Fix: Always check the base of the logarithm and ensure you are using the correct one in your calculations.
Forgetting to simplify intermediate steps before applying the laws
Why it happens: Students sometimes rush through problems and skip simplifying intermediate steps, leading to errors.
Fix: Take the time to simplify each step before moving on to the next one.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Simplify Logarithms | Use the laws of logarithms to combine and evaluate a logarithmic expression. | 3 |
| Simplify Logarithms | Use the laws of logarithms to simplify a logarithmic expression to a single value. | 3 |
| Simplify Logarithms | Apply the power law of logarithms to simplify log base 3 of 9 to the fourth power. | 3 |
| Simplify Logarithms | Apply the laws of logarithms to rewrite and simplify a logarithmic expression. | 3 |
| Total across these question types | 12 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.