GCSE Further Maths · Topic 6
Further topics
These topics complete the Additional Maths toolkit.
We will use logarithms and exponentials, the binomial expansion, and permutations and combinations.
Then we finish with vectors and arithmetic and geometric progressions.
- 5 subtopics
- 209 questions
- Free: 6.1 notes and 10 questions
In this topic
- 6.1Logarithms and exponentials: laws of logs, solving equations and eFree below · 35 questions
- 6.2The binomial expansion: Pascal's triangle and nCrOpen in Full Marks · 48 questions
- 6.3Permutations and combinations: arrangements and selectionsOpen in Full Marks · 52 questions
- 6.4Vectors in two dimensions: position vectors, magnitude and velocityOpen in Full Marks · 21 questions
- 6.5Arithmetic and geometric progressions and their sumsOpen in Full Marks · 53 questions
6.1 · Free notes
Logarithms and exponentials: laws of logs, solving equations and e
Key points
- Definition: y = aˣ ⇔ x = logₐ y. For example log₂ 8 = 3 because 2³ = 8. logₐ a = 1 and logₐ 1 = 0.
- Laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x.
- Solving aˣ = b: take logs, x = log b / log a. For example 3ˣ = 20 gives x = log 20 / log 3 ≈ 2.73.
- The number e ≈ 2.718 gives the exponential function eˣ; its inverse is the natural logarithm ln x. eˣ models continuous growth and decay, such as N = N₀e^(−kt).
- Graphs: y = aˣ (a > 1) passes through (0, 1) and grows quickly; y = logₐ x passes through (1, 0); they are reflections of each other in y = x.
In more detail
- Worked example: solve log x + log(x − 3) = 1 (base 10). log(x(x − 3)) = 1, so x² − 3x = 10, (x − 5)(x + 2) = 0, x = 5 (x = −2 is rejected since log of a negative is undefined).
- Change of base: logₐ b = log b / log a, useful when your calculator only has log and ln.
- Exam tip: check solutions in the original equation; logarithms of zero or negative numbers do not exist.
Practice
Logarithms and exponentials: laws of logs, solving equations and e: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is log₂ 32?
- 16
- 5
- 4
- 6
Show answer
B: 5. 2⁵ = 32.
2. Write log 3 + log 4 as a single logarithm.
- log 7
- log 3/4
- log 81
- log 12
Show answer
D: log 12. log x + log y = log(xy).
3. Simplify 3 log 2.
- log 6
- log 5
- log 8
- log 9
Show answer
C: log 8. log 2³ = log 8.
4. Solve 2ˣ = 10 to 3 s.f.
- 3.32
- 5
- 0.301
- 20
Show answer
A: 3.32. x = log 10 / log 2 ≈ 3.32.
5. What is ln(e⁴)?
- e⁴
- 4
- 1
- 4e
Show answer
B: 4. ln and e are inverses.
6. What is logₐ 1 for any valid base a?
- 1
- a
- Undefined
- 0
Show answer
D: 0. a⁰ = 1.
7. Solve log₃ x = 4.
- 12
- 64
- 81
- 7
Show answer
C: 81. x = 3⁴.
8. Which point does y = 5ˣ always pass through?
- (0, 1)
- (1, 0)
- (0, 5)
- (5, 1)
Show answer
A: (0, 1). 5⁰ = 1.
9. Simplify log 20 − log 5.
- log 15
- log 4
- log 100
- 4
Show answer
B: log 4. log(20/5).
10. A population P = 100e^(0.2t). What is P when t = 0?
- 0
- 120
- 20
- 100
Show answer
D: 100. e⁰ = 1.
Keep going in Full Marks
Full Marks has 35 questions on Logarithms and exponentials: laws of logs, solving equations and e (25 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Further topics, plus games, timed exams and past papers.
On your exam board
- OCR ATopic 2: Enumeration and probability
- CambridgeTopic 6: Logarithmic and exponential functions, Topic 11: Permutations and combinations, Topic 12: Series, Topic 13: Vectors in two dimensions
Not in the AQA specification.