Expressing one log in terms of others

A recurring Merit/Excellence task: given log_a of two numbers, express log_a of a THIRD, related number without a calculator.

The method

Write the target number as a PRODUCT, QUOTIENT, or POWER of the numbers you already have logs for — e.g. 75 = 3 × 25 = 3 × 5², so log_a 75 can be built from log_a 3 and log_a 5.
Apply the log laws to split the log of that product/quotient/power into a combination of the known logs: log_a(3 × 5²) = log_a 3 + 2·log_a 5.
Substitute the given values (p, q, or similar) for each known log to get the final answer purely in terms of those letters.
The same idea extends to "change of base" problems — if log_b x = 2, then b² = x, and this relationship can be substituted into a second equation involving x or b.

Worked example Writing log_a 75 in terms of p and q Given log_a 3 = p and log_a 5 = q, write log_a 75 in terms of p and q. Factorise 75 into numbers matching the given logs: 75 = 3 × 25 = 3 × 5². Apply the product law: log_a 75 = log_a(3 × 5²) = log_a 3 + log_a(5²). Apply the power law to the second term: log_a(5²) = 2·log_a 5. Substitute: log_a 75 = p + 2q. ✓ The whole method hinges on finding the right factorisation of the target number FIRST — 75 = 3 × 5² is the only factorisation that uses exactly the two given logs. ✕

Trying to combine logs of numbers that share no factors with what is given

This technique only works when the target number can be built ENTIRELY from products/quotients/powers of the numbers already given logs for. If a problem gives log_a 3 and log_a 5 but asks for log_a 7, there is no valid factorisation — check the target number factorises into exactly the given bases before attempting this method.

Log law	When to reach for it
log(ab) = log a + log b	Target number is a PRODUCT of the given numbers
log(a/b) = log a − log b	Target number is a QUOTIENT of the given numbers
log(aⁿ) = n·log a	Target number is a POWER of a given number

Practice for this standard covers the whole topic — flashcards and quiz.