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 log2 8 mean?
Step by step
- Ask: what power of 2 makes 8?
- Since 23 = 8, the logarithm equals 3.
Final answer
log2 8 = 3
Logarithm = exponent: That is the whole idea to keep returning to: a logarithm gives you an exponent.
| 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
log5 125 = 3
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?