Indices & Logarithms

21 Grade 9 Mathematics questions on Indices & Logarithms, with worked marking schemes. 21 questions are available on this topic in full.

1. Question 1

   1 mark

   What is log₁₀ 100?

   A. 1

   B. 100

   C. 2

   D. 10

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

2. Question 2

   2 marks

   Evaluate without a calculator: log₁₀ 10,000 = ___

   (2 marks)

   Marking scheme for this question

   Create a free account to see it

3. Question 3

   2 marks

   Express 0.001 as a power of 10. Then find its logarithm.

   (2 marks)

   Marking scheme

   $0.001$ is $\frac{1}{1000}$, and $1000=10^3$, so $0.001=10^{-3}$. A negative index means one over that power. (1)

   The logarithm is the index itself, so $\log 0.001=-3$. (1)

   Final answer: $0.001=10^{-3}$ and $\log 0.001=-3$.

4. Question 4

   1 mark

   Simplify: 6⁴ ÷ 6⁴ = ___

   A. 0

   B. 6⁸

   C. 6

   D. 1

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

5. Question 5

   1 mark

   Simplify: 3⁶ ÷ 3² = ___

   A. 3³

   B. 3⁴

   C. 3⁸

   D. 1

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

6. Question 6

   1 mark

   Express 8 as a power of 2.

   A. 2²

   B. 2³

   C. 2⁸

   D. 2⁴

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

7. Question 7

   1 mark

   If log x = 2, then x = ___

   A. 20

   B. 100

   C. 2

   D. 10

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

8. Question 8

   3 marks

   Simplify, giving your answer as a single power: 2⁸ × 2³ ÷ 2⁵

   (3 marks)

   Marking scheme

   When multiplying powers of the same base, add the indices; when dividing, subtract. (1)

   So $2^8\times 2^3÷2^5=2^{8+3-5}$. (1)

   $8+3-5=6$, giving $2^6$. (1)

   Final answer: $2^6$, which is $64$.

9. Question 9

   1 mark

   Evaluate: (2³)² = ___

   A. 2⁹

   B. 2⁵

   C. 4⁶

   D. 2⁶

   (1 mark)

   Marking scheme for this question

   Create a free account to see it

10. Question 10

    1 mark

    Simplify: x⁵ × x³ ÷ x⁴ = ___

    A. x¹²

    B. x⁶

    C. x⁴

    D. x²

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

11. Question 11

    1 mark

    log₂ 8 = ___

    A. 8

    B. 4

    C. 2

    D. 3

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

12. Question 12

    1 mark

    Simplify: 2³ × 2⁴ = ___

    A. 2¹

    B. 2⁷

    C. 2¹²

    D. 4⁷

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

13. Question 13

    1 mark

    Evaluate: 5⁰ = ___

    A. 1

    B. 5

    C. ∞

    D. 0

    (1 mark)

    Marking scheme

    A. $1$ — anything (except zero) raised to the power zero is $1$. It follows from dividing a power by itself: $\frac{5^n}{5^n}=5^0$, and that is clearly $1$. D is the commonest confusion.

14. Question 14

    1 mark

    Simplify: (4²)³ = ___

    A. 8⁶

    B. 4⁶

    C. 4⁵

    D. 4²

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

15. Question 15

    1 mark

    Evaluate: 4^(1/2) = ___

    A. 1

    B. 8

    C. 16

    D. 2

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

16. Question 16

    1 mark

    log 1,000 = ___

    A. 2

    B. 1

    C. 4

    D. 3

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

17. Question 17

    2 marks

    Solve for n: 3ⁿ = 81

    (2 marks)

    Marking scheme for this question

    Create a free account to see it

18. Question 18

    1 mark

    log 1 = ___

    A. 0

    B. 1

    C. −1

    D. undefined

    (1 mark)

    Marking scheme for this question

    Create a free account to see it

19. Question 19

    2 marks

    Evaluate ${\log }_{2}8+{\log }_{3}9$.

    (2 marks)

    Marking scheme for this question

    Create a free account to see it

20. Question 20

    1 mark

    Evaluate: 2⁻³ = ___

    A. $\frac{1}{8}$

    B. $\frac{1}{6}$

    C. $\frac{1}{2}$

    D. −8

    (1 mark)

    Marking scheme

    A. $\frac{1}{8}$ — a negative index means one over the positive power: $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$. D treats the minus as making the answer negative, which it never does.

21. Question 21

    3 marks

    Given that log 3 ≈ 0.4771, evaluate log 27.

    (3 marks)

    Marking scheme

    Write $27$ as a power of $3$: $27=3^3$. (1)

    The power rule for logarithms brings the index to the front: $\log 3^3=3\log 3$. (1)

    Substitute $\log 3\approx 0.4771$: $3\times 0.4771=1.4313$. (1)

    Final answer: $\log 27\approx 1.4313$.

22. 16 more marking schemes are on this page. A free account opens all of them, and lets you build your own paper on the topics you have actually taught, laid out in KNEC format with its full marking scheme.

    Create a free account

More Grade 9 Mathematics topics