Logarithms — Grade 12 Maths (CAPS)

Logs as the reverse of exponents — evaluating, converting between log and exponent form, solving for an unknown exponent such as the n in a finance question, and the graph of y = log_b x as the inverse of y = b^x — each worked in full, with the log laws as enrichment and a worksheet.

A logarithm answers one question: 'what power do I raise the base to, to get this number?' It's the exact reverse of an exponent — that single idea makes logs easy.

if b^x = y then log_b y = x
for b > 0, b ≠ 1 and y > 0

The conditions that come with it: the base must be positive and not 1 (b > 0, b ≠ 1), and you may only take the log of a POSITIVE number (y > 0). That is why log 0 and log(−5) do not exist, and why a log equation's answers must always be checked against the original expression.

Read log₂ 8 as 'the power that turns 2 into 8'. Since 2³ = 8, log₂ 8 = 3.

1Type 1: Evaluate a logarithm

Worked ExampleExample 1: evaluate log₅ 125

1
Ask: 5 to what power gives 125?
The meaning of the log.
2
5³ = 125.
5 × 5 × 5 = 125.
3
So log₅ 125 = 3.
The logarithm is the exponent.

2Type 2: Convert between log and exponent form

Worked ExampleExample 2: write 2⁴ = 16 in log form, and log₃ 81 = 4 in exponent form

1
2⁴ = 16 → log₂ 16 = 4.
Base stays the base; the exponent becomes the log's value.
2
log₃ 81 = 4 → 3⁴ = 81.
Reverse the same relationship.

3Beyond the ATP (not examined): the log laws

The Grade 12 ATP examines the definition of a logarithm, its graph and its use in finding n; it does not examine the log laws or change of base. They are here because they explain how a calculator evaluates logs — read them, but do not expect them in the exam.

log(xy) = log x + log y • logx/y = log x − log y • log xⁿ = n log x (x, y > 0)

Worked ExampleExample 3: simplify log 4 + log 25

1
Product law: log 4 + log 25 = log(4 × 25) = log 100.
Adding logs → multiply the numbers.
2
log₁₀ 100 = 2.
10² = 100.

4Beyond the ATP (not examined): change of base

The Casio fx-82ZA PLUS used in South African schools has a log□ key for any base. On a calculator with only log (base 10) and ln keys, use the change-of-base rule: log_b y = (log y)/(log b).

Worked ExampleExample 4: evaluate log₂ 20

1
log₂ 20 = log 20 ÷ log 2.
Change-of-base rule (any base on the calculator).
2
≈ 1,301 ÷ 0,301 ≈ 4,32.
Use the log button.

5Type 3: Solve for an unknown exponent

Worked ExampleExample 5: solve 3ˣ = 40

1
Write it in log form: x = log₃ 40.
By definition, 3^x = 40 means x is the power that turns 3 into 40.
2
Evaluate on the calculator with the log□ key: x ≈ 3,36.
Check: 3^3,36 ≈ 40 ✓

💡

Logs let you solve for an unknown in the exponent — e.g. 'how long until an investment doubles?' Write the equation in log form and evaluate it. Logs are also the inverse of the exponential graph.

⚠️

log(a + b) is NOT log a + log b. The laws work on products and quotients — log(ab) = log a + log b — never on sums. One of the most common log errors.

Logs appear in finance, functions and exponential equations across matric. Master these five types in the worksheet.

y = log₂ x is the mirror image of y = 2^x in the line y = x — which is exactly what 'inverse' means.

6The graph when the base is between 0 and 1

y = log₂ x rises; y = log_½ x falls. Both pass through (1 ; 0) and both have the y-axis as asymptote.

If b > 1 the log graph increases. If 0 < b < 1 it decreases — it is the first graph reflected in the x-axis. Either way the domain is x > 0, the range is y ∈ ℝ, and the y-axis is a vertical asymptote.

⚠️

log_½ 8 = −3, not 3. A base below 1 gives negative outputs for x > 1 — check the sign against the graph before you write it down.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Evaluate log₂ 16. (evaluate)
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2⁴ = 16 → 4
2
Evaluate log₃ 27. (evaluate)
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3³ = 27 → 3
3
Evaluate log₁₀ 1 000. (evaluate)
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10³ = 1000 → 3
4
Write 5³ = 125 in log form. (convert)
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log₅ 125 = 3
5
Write log₂ 32 = 5 in exponent form. (convert)
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2⁵ = 32
6
Solve log₂ x = 5. (definition)
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x = 2⁵ (1)
x = 32 (1)
7
Write down the inverse of f(x) = 3^x in the form y = … (inverse)
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swap x and y: x = 3^y (1)
y = log₃ x (1)
8
State the domain and the range of y = log₂ x. (graph)
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domain: x > 0 (1)
range: y ∈ ℝ (1)
9
Money invested at 8% p.a. compound interest doubles after n years. Find n (2 dp). (finance)
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2 = (1,08)ⁿ (1)
n = log_1,08 2 ≈ 9,01 years (1)
10
Solve 2^x = 20 (2 dp). (solve)
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x = log₂ 20 (2)
x ≈ 4,32 (1)