Logarithms: The Complete Grade 12 Guide — Worksheet & Memo

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Grade 12 · Functions

Logarithms: The Complete Grade 12 Guide

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 36

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
log₂8 = …
(1)
A)3✓
B)4
C)16
D)2
Answer: A — 2³ = 8.
- B — 2⁴ = 16, not 8
- C — multiplied 2 by 8
- D — read off the base instead of the exponent
1.2
log 100 = …
(1)
A)2✓
B)10
C)100
D)1 000
Answer: A — The base is 10, and 10² = 100.
- B — read off the base instead of the exponent
- C — read off the argument instead of the exponent
- D — 10³ = 1 000, not 100
1.3
log a + log b = …
(1)
A)log(ab)✓
B)log(a + b)
C)log a × log b
D)log alog b
Answer: A — Adding two logarithms MULTIPLIES their arguments.
- B — adding the logs multiplies the arguments; it does not add them
- C — the logarithms are added, not multiplied
- D — that would come from log a − log b
1.4
log 200 − log 2 = …
(1)
A)2✓
B)log 198
C)100
D)1
Answer: A — log 200 − log 2 = log(200 ÷ 2) = log 100 = 2.
- B — subtracting logarithms DIVIDES the arguments; it does not subtract them
- C — that is 200 ÷ 2, before the logarithm is taken
- D — log 100 = 2, not 1
1.5
log(x³) = …
(1)
A)3 log x✓
B)(log x)³
C)log 3x
D)log x3
Answer: A — A power inside a logarithm comes out in front as a multiplier.
- B — the 3 comes OUT of the logarithm; it does not stay as a power of the whole log
- C — the 3 was a power of x, not a multiplier of x
- D — the power comes out as a multiplier, not as a divisor
1.6
If log₃x = 4, then x = …
(1)
A)81✓
B)12
C)64
D)1,32
Answer: A — log₃x = 4 means 3⁴ = x.
- B — multiplied 3 by 4
- C — used 4³ instead of 3⁴
- D — took the fourth root of 3 instead of the fourth power
1.7
log 1 = …
(1)
A)0✓
B)1
C)10
D)undefined
Answer: A — Any base raised to the power 0 gives 1.
- B — log 10 = 1; it is log 1 that is 0
- C — that is the base, not the answer
- D — log 1 is perfectly well defined; it is log 0 that is not
1.8
log(−5) is …
(1)
A)undefined✓
B)−log 5
C)0
D)5
Answer: A — No power of a positive base gives a negative result, so the logarithm of a negative number does not exist.
- B — a minus cannot be pulled out of a logarithm's argument
- C — log 1 = 0, but −5 is not 1
- D — that is the argument itself
1.9
Using the change-of-base rule, log₅20 = …
(1)
A)log 20log 5✓
B)log 5log 20
C)log 20 − log 5
D)log 4
Answer: A — log_ba = log a ÷ log b, in any convenient base.
- B — the NEW base goes on the bottom
- C — that is log(20 ÷ 5), a different quantity
- D — log(20 ÷ 5) = log 4, which is not log₅20
1.10
Solve for x, correct to TWO decimal places: 2^x = 20
(1)
A)4,32✓
B)10
C)1,30
D)0,30
Answer: A — x = log 20 ÷ log 2 = 4,32.
- B — halved 20 instead of solving with logarithms
- C — that is log 20 on its own
- D — that is log 2 on its own

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[13 MARKS]

Solve for x. Show ALL your working.

2.1
log₂ (x + 3) + log₂ (x − 3) = 4
(6)
Combine the logs: log₂[(x + 3)(x − 3)] = 4 (1)
x² − 9 = 2⁴ (2)
x² = 25 (1)
x = ±5 (1)
x = −5 is REJECTED — log₂(−2) does not exist, so x = 5 (1)
2.2
3^x = 20, correct to TWO decimal places
(4)
Take logs of both sides: x log 3 = log 20 (2)
x = log 20log 3 (1) = 2,73 (1)
2.3
log_x 81 = 4
(3)
x⁴ = 81 (2)
x = 3 (1) — a log base must be positive

Question 3

[13 MARKS]

Answer the questions below WITHOUT the use of a calculator.

3.1
Simplify: 2 log 5 + log 4
(4)
= log 5² + log 4 (1)
= log(25 × 4) (2)
= log 100 = 2 (1)
3.2
Simplify: log 50 − log 5
(3)
= log(505) (2) = log 10 = 1 (1)
3.3
Write down the equation of the inverse of f(x) = 2^x in the form y = …
(3)
Swap x and y: x = 2^y (2)
y = log₂ x (1)
3.4
Write down the domain of y = log₂ x, and explain why.
(3)
x > 0 (2)
No power of 2 gives zero or a negative number, so the log of such a number does not exist (1)

TOTAL: 36 marks

This question paper consists of 3 questions.