Logarithms • Topic 1 of 4

Log Laws

Everything in this chapter flows from three laws that convert harder operations into easier ones. The product law log_b(MN) = log_b M + log_b N turns multiplication into addition; the quotient law log_b(M/N) = log_b M − log_b N turns division into subtraction; and the power law log_b(M^n) = n·log_b M pulls an exponent out in front. Add two anchors — log_b b = 1 and log_b 1 = 0 — and most expression-simplification questions collapse in two lines. The CAT skill is reading a messy expression and spotting which law to apply: a coefficient in front of a log is a hidden power (3·log 2 = log 8), and a sum of logs with the same base is a single log of a product. Always keep the base the same before combining; you can only merge logs that share a base.

✅ Solved examples

1. Simplify log 2 + log 5 (base 10).
Product law: log(2×5) = log 10 = 1.
2. Express 2·log 3 + log 5 − log 9 as a single log (base 10).
2·log 3 = log 9. So log 9 + log 5 − log 9 = log 5 ≈ 0.699.
3. If log 2 = 0.301, find log 8.
log 8 = log 2^3 = 3·log 2 = 3 × 0.301 = 0.903.
4. Simplify log_3 81 − log_3 9.
log_3 81 = 4, log_3 9 = 2 (since 3^4 = 81, 3^2 = 9). 4 − 2 = 2. Check: log_3(81/9) = log_3 9 = 2.

✏️ Practice — try these, take hints as needed

1. Simplify log 4 + log 25 (base 10).
Combine into one log via the product law.
log(4 × 25).
log 100 = 2.
2
2. Write log 12 in terms of log 2 and log 3.
12 = 2^2 × 3.
log(2^2 × 3) = log 2^2 + log 3.
Apply the power law to log 2^2.
2·log 2 + log 3
3. If log 3 = 0.477, find log 27.
27 = 3^3.
log 3^3 = 3·log 3.
3 × 0.477.
1.431
4. Simplify log_5 1000 − log_5 8.
Quotient law first.
log_5(1000/8) = log_5 125.
125 = 5^3.
3
5. Simplify ½·log 36 + log 5 − log 3 (base 10).
½·log 36 = log 36^(1/2) = log 6.
log 6 + log 5 − log 3.
log(6×5/3) = log 10.
1

📝 Topic test — 8 questions

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