Logarithms Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Logarithms questions. See exactly how to solve problems on logarithm as inverse, evaluate, base 10, log as inverse.

logarithm as inverseevaluatebase 10log as inverseconverting formnegative index
A-Level70 questionsStep-by-step solutions
Question 1
1 markeasy
Evaluate log⁡28\log_2 8.

Worked solution

  1. Rewrite the log as a power question

    log⁡28=?  ⇒  2?=8\log_2 8 = ? \;\Rightarrow\; 2^{?} = 8

    A logarithm just asks 'what power do I raise the base to?' Here the base is 2, so we ask what power of 2 gives 8. This is the inverse of raising 2 to a power.

  2. Write 8 as a power of 2

    8=238 = 2^3

    We list powers of 2: 2, 4, 8. So 8 is the third one, meaning 8 = 2 cubed. Recognising powers of small numbers is a key skill from indices.

  3. Read off the exponent

    23=8  ⇒  log⁡28=32^3 = 8 \;\Rightarrow\; \log_2 8 = 3

    Since 2 to the power 3 equals 8, the logarithm is exactly that power, 3. So the answer is 3.

Answer
log⁡28=3\log_2 8 = 3
Question 2
1 markeasy
Evaluate log⁡101000\log_{10} 1000.

Worked solution

  1. Turn the log into a power question

    log⁡101000=?  ⇒  10?=1000\log_{10}1000 = ? \;\Rightarrow\; 10^{?} = 1000

    log base 10 asks what power of 10 gives the number. Base-10 logs are the ones your calculator's 'log' button uses.

  2. Write 1000 as a power of 10

    1000=1031000 = 10^3

    1000 is 10 x 10 x 10, which is 10 cubed. Counting the zeros (three of them) also gives the power.

  3. State the answer

    log⁡101000=3\log_{10}1000 = 3

    The power needed is 3, so the logarithm is 3.

Answer
log⁡101000=3\log_{10}1000 = 3
Question 3
1 markeasy
Evaluate log⁡525\log_5 25.

Worked solution

  1. Rewrite as a power question

    5?=255^{?} = 25

    We ask what power of 5 gives 25. The base of the logarithm becomes the base of the power.

  2. Express 25 as a power of 5

    25=5225 = 5^2

    5 times 5 is 25, so 25 is 5 squared.

  3. State the answer

    log⁡525=2\log_5 25 = 2

    The required power is 2, so the logarithm equals 2.

Answer
log⁡525=2\log_5 25 = 2
Question 4
2 markseasy
Write 25=322^5 = 32 in logarithmic form.

Worked solution

  1. Recall the link between powers and logs

    ax=b  ⟺  log⁡ab=xa^x = b \;\Longleftrightarrow\; \log_a b = x

    Every index statement can be rewritten as a logarithm statement and vice versa. The base of the power becomes the base of the log.

  2. Identify the parts

    a=2,x=5,b=32a = 2,\quad x = 5,\quad b = 32

    Match the numbers: the base is 2, the power is 5, and the result is 32.

  3. Write it in log form

    log⁡232=5\log_2 32 = 5

    Put the base as the little number, the result inside, and the power on the right. This says 2 raised to the power 5 is 32.

Answer
log⁡232=5\log_2 32 = 5
Question 5
2 markseasy
Write log⁡381=4\log_3 81 = 4 in exponential (index) form.

Worked solution

  1. Recall the link between logs and powers

    log⁡ab=x  ⟺  ax=b\log_a b = x \;\Longleftrightarrow\; a^x = b

    A logarithm equation can always be turned back into an index equation. The base stays the base.

  2. Identify the parts

    a=3,x=4,b=81a = 3,\quad x = 4,\quad b = 81

    The base of the log is 3, the value it equals is 4, and the number inside is 81.

  3. Write it in index form

    34=813^4 = 81

    So 3 raised to the power 4 gives 81. You can check: 3x3x3x3 = 81.

Answer
34=813^4 = 81

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