If log 3 = 0.4771 and log 2 = 0.3010, what is the value of log 12?

Analyze the question: We are given log 3 and log 2 and asked to find log 12. The trick is to write 12 using only the numbers 2 and 3.

Concept: Two log laws are needed.

• Product law: log(ab) = log a + log b
• Power law: log(aⁿ) = n log a

Solution:

12 = 2 × 2 × 3 = 2² × 3

log 12 = log(2² × 3) = log 2² + log 3 = 2 log 2 + log 3 = 2(0.3010) + 0.4771 = 0.6020 + 0.4771 = 1.0791

Check against the options:

• A (0.0432), B (0.4322), D (1.2552) and E (1.5225) do not match.
• C (1.

Sense check: log 10 = 1 and log 100 = 2. Since 12 lies between 10 and 100, log 12 must lie between 1 and 2, so 1.0791 is sensible. It should also be just above 1, because 12 is only slightly above 10.

Exam tip: When you are asked for the log of a number that isn't given, break it into prime factors (12 = 2 × 2 × 3) and add their logs. Here, log 12 = log 2 + log 2 + log 3. This avoids mistakes with the power law.

Final answer: C