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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Apply the product rule: log 2 (32 × 8) = log 2 32 + log 2 8
- S2
Calculate each logarithm: log 2 32 = 5 and log 2 8 = 3
- S3
Add the results: 5 + 3 = 8
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Apply the product rule: log 2 (32 × 8) = log 2 32 + log 2 8
- S2
Calculate each logarithm: log 2 32 = 5 and log 2 8 = 3
- S3
Add the results: 5 + 3 = 8
Final answer: 8
Common mistakes
- Forgetting to apply the product rule when simplifying log a (xy) — 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) — 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 ) — 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 — 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 — 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 — Always remember to use the correct form: log a (x -1 ) = -log a x
- Using the wrong base for logarithms in calculations — 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 — Take the time to simplify each step before moving on to the next one.
Where the marks go
- Full worked solution (all marking points)3 marks