Logarithms • Topic 2 of 4

Change of Base

When a problem mixes logs of different bases — say log_2 7 and log_3 7 — you cannot combine them until they share a base. The change-of-base rule fixes this: log_b a = (log_c a)/(log_c b) for any valid base c. Pick c = 10 or whatever simplifies the numbers. Two consequences are CAT favourites. First, the reciprocal rule: log_b a · log_a b = 1, so log_b a = 1/log_a b — flipping a log inverts its base and argument. Second, telescoping chains: log_2 3 · log_3 4 · log_4 5 · … · log_(n−1) n collapses to log_2 n, because each fraction cancels with the next. Also remember the base-power rule log_(b^m)(a^n) = (n/m)·log_b a, which lets you slide powers between the base and the argument. Spotting these collapses turns a scary product of logs into a one-line answer.

✅ Solved examples

1. Evaluate log_8 32.
log_8 32 = (log_2 32)/(log_2 8) = 5/3. (Or base-power rule: log_(2^3) 2^5 = 5/3.)
2. If log_2 3 = 1.585, find log_3 2.
Reciprocal rule: log_3 2 = 1/log_2 3 = 1/1.585 ≈ 0.631.
3. Simplify log_2 3 · log_3 4 · log_4 5 · log_5 8.
Telescopes to log_2 8 = 3. (Each link cancels: 3 → 4 → 5 → 8 in numerator/denominator chain.)
4. Evaluate log_9 27.
log_9 27 = (log_3 27)/(log_3 9) = 3/2. (Base-power: log_(3^2) 3^3 = 3/2.)

✏️ Practice — try these, take hints as needed

1. Evaluate log_4 8.
Convert to base 2.
(log_2 8)/(log_2 4).
3/2.
3/2
2. If log_5 7 = 1.209, find log_7 5.
Use the reciprocal rule.
log_7 5 = 1/log_5 7.
1/1.209.
≈ 0.827
3. Simplify log_3 5 · log_5 9.
Change both to a common base.
log_3 5 · log_5 9 = log_3 9.
9 = 3^2.
2
4. Evaluate log_27 81.
Both are powers of 3.
log_(3^3) 3^4 = 4/3.
Use the base-power rule.
4/3
5. Simplify log_2 6 · log_6 8.
Chain rule cancels the 6.
= log_2 8.
8 = 2^3.
3

📝 Topic test — 8 questions

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