The big idea: A logarithm asks: what power must I raise the base to, to get the given number?
- base
- result
- required power
Worked example
What does log_2 8 mean?
Step by step
- Ask: what power of 2 makes 8?
- Since 23 = 8, the logarithm equals 3.
Final answer
Logarithm = exponent: That is the whole idea to keep returning to: a logarithm gives you an exponent.
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| Exponential form | Logarithmic form |
|---|---|
| 102 = 100 | log10 100 = 2 |
| 34 = 81 | log3 81 = 4 |
| 25 = 32 | log2 32 = 5 |
Worked example
Rewrite 53 = 125 in logarithmic form.
Step by step
- Base stays 5, answer stays 125, power becomes the logarithm result.
Final answer
Do not swap the numbers randomly: The base stays the base.
The result stays the result.
The exponent becomes the value of the logarithm.
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Worked examples
Evaluate (a) log10 1000 (b) log4 16 (c) log7 1
Step by step
- 103 = 1000, so log10 1000 = 3.
- 42 = 16, so log4 16 = 2.
- Any non-zero base to the power 0 gives 1.
Final answer
3, 2, and 0
Think in powers: Do not try to memorize separate log facts.
Rewrite the question as a power statement and ask what exponent is needed.
Base 10 is special: When you see just log x on a calculator or in many IB questions, it usually means log base 10.
| Notation | Meaning |
|---|---|
| log 100 | log10 100 |
| log 1000 | log10 1000 |
| ln x | log base e of x (later work) |
Worked example
Evaluate log 10000.
Step by step
- This means log base 10.
- Since 104 = 10000, the answer is 4.
Final answer
4
Do not ignore the base idea: Even when the base is not written, the same question is still being asked: what power of 10 gives this number?
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Two GDC keys for log questions:
- LOG → evaluates log₁₀ (use when finding the dB / pH value).
- 2nd + LOG → 10ˣ (use to undo a log and find the original value).
✏️ IB-style worked examples
Part (i) — find sound level from intensity
Sound level L (in dB) and intensity I (in W m⁻²) are linked by L = 10 log₁₀(I × 10¹²).
A rehearsal room has intensity I = 2.5 × 10⁻⁴ W m⁻². Find the sound level.
Step by step
- Plug in I = 2.5 × 10⁻⁴. Replace I in the model with the given intensity.
- Combine the powers of 10 using , so :
- Type it into the GDC using LOG for log₁₀ and EE (2nd + ,) for the ×10⁸. Then round to 3 s.f.:
Final answer
L ≈ 84.0 dB
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (ii) — find intensity from sound level
Using the same model L = 10 log₁₀(I × 10¹²):
A loud concert has sound level L = 95 dB. Find the intensity I.
Step by step
- Set L = 95. Put the given sound level into the model — now I is the unknown:
- Get rid of the 10 in front of the log. Divide both sides by 10 so we have only a log on one side:
- Undo the log with 10ˣ. Because log₁₀ and 10ˣ are inverses, raising 10 to both sides peels the log away:
- Isolate I. Divide both sides by 10¹² using :
- Convert to standard form and round to 3 s.f.:
Final answer
I ≈ 3.16 × 10⁻³ W m⁻²
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.