The big idea: Logarithms turn multiplication into addition. Three laws (same base throughout) let you combine logs into one or break them apart.
See it on numbers
Show that log₂8 + log₂4 = log₂32.
Step by step
- Product law: adding logs multiplies the arguments.
- Work out the inside.
Final answer
Both sides equal 5.
The #1 trap: log(x + y) is NOT log x + log y. The laws act on products, quotients and powers — never on a sum.
Two handy values: loga 1 = 0 and loga a = 1.
Read the laws right-to-left: A common Paper 1 opener gives a sum or multiple of logs and asks for it as one logarithm — a coefficient becomes a power, + becomes ×, − becomes ÷.
IB-style question — write as a single logarithm
Write ln 6 + 2 ln 3 − ln 2 as a single logarithm.
Step by step
- Coefficient first: the power law moves the 2 up.
- Add → multiply, subtract → divide.
- Simplify the inside.
Final answer
ln 27.
IB-style question — with variables
Write 3 log x + log y − 2 log z as a single logarithm.
Step by step
- Move each coefficient up as a power.
- Add → multiply on top, subtract → divide.
Final answer
log(x³y / z²).
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Break it apart: The reverse skill: split one logarithm into pieces, or write it using given logs — factorise the inside, then apply the product and power laws.
IB-style question — expand a single log
Write log₂(8x³) as a sum of simpler terms.
Step by step
- Product law splits the multiplication.
- Evaluate log₂8, and bring the power down.
Final answer
3 + 3 log₂ x.
IB-style question — in terms of given logs
Given that log 2 = p and log 3 = q (base 10), write log 24 in terms of p and q.
Step by step
- Factorise the inside into 2s and 3s.
- Split with the product law, then bring the power down.
- Replace with the given letters.
Final answer
3p + q.
Switch to a base you can handle: When the base is awkward, change it: rewrite loga x as a ratio of logs in any base you choose — then evaluate.
IB-style question — evaluate a logarithm
Evaluate log₈ 32 without a calculator.
Step by step
- Both numbers are powers of 2 — change to base 2.
- Evaluate the top and bottom.
Final answer
5/3.
IB-style question — change of base with given logs
Given that log 2 = p and log 3 = q (base 10), write log₃ 8 in terms of p and q.
Step by step
- Change to base 10.
- Write 8 = 2³ and bring the power down.
- Replace with the given letters.
Final answer
3p / q.