AP PRECALCULUS — UNIT 2B · EXPONENTIAL & LOGARITHMIC FUNCTIONS
2.12 — Logarithmic Function Manipulation
Notes — Properties and the Change-of-Base Formula
💡 Learning Objectives (2.12.A)
By the end of this lesson you will be able to:
- Apply the product, quotient, and power properties of logarithms to rewrite expressions
- Expand a single logarithm into a sum or difference of simpler logarithms
- Condense an expression of multiple logarithms into a single logarithm
- Use the change-of-base formula to evaluate logarithms with any base
1. The Three Core Log Properties
All three properties come straight from the corresponding rules for exponents and the inverse relationship from 2.10. Throughout, b > 0, b ≠ 1, and the arguments are positive.
💡 Properties of Logarithms
- Product: log_b(MN) = log_b(M) + log_b(N)
- Quotient: log_b(M/N) = log_b(M) − log_b(N)
- Power: log_b(M^k) = k · log_b(M)
⚠️ Common mistakes
- log(M + N) is NOT log(M) + log(N). Logs distribute over multiplication, not addition.
- (log_b(M))^k is NOT k · log_b(M). The power rule applies only when the exponent is on the argument.
- log(M)/log(N) is NOT log(M/N). The quotient rule says log(M/N) = log(M) − log(N), with subtraction.
2. Expanding Logarithms
‘Expanding’ a logarithm means using the three properties to break a single complicated log into a sum or difference of simpler ones.
📘 Example — Expand log₂(8x³y / z⁵)
- Use the quotient rule: log₂(8x³y) − log₂(z⁵)
- Use the product rule on the first piece: log₂(8) + log₂(x³) + log₂(y) − log₂(z⁵)
- Use the power rule on x³ and z⁵: log₂(8) + 3 log₂(x) + log₂(y) − 5 log₂(z)
- Evaluate log₂(8) = 3: 3 + 3 log₂(x) + log₂(y) − 5 log₂(z)
3. Condensing Logarithms
‘Condensing’ runs the same properties in reverse — combine multiple log terms into a single log.
📘 Example — Condense 2 ln(x) + ln(y) − (1/2) ln(z)
- Apply power rule to each term: ln(x²) + ln(y) − ln(z^(1/2))
- Apply product rule to the first two: ln(x²y) − ln(√z)
- Apply quotient rule: ln(x²y / √z)
4. The Change-of-Base Formula
Calculators only have ‘log’ (base 10) and ‘ln’ (base e). To evaluate something like log₇(50), use the change-of-base formula:
log_b(x) = log_a(x) / log_a(b)
The new base a can be anything that makes calculation possible — usually 10 or e.
📘 Example — Evaluate log₇(50)
- Choose base 10: log₇(50) = log(50) / log(7)
- Compute: log(50) ≈ 1.69897, log(7) ≈ 0.84510
- Divide: log₇(50) ≈ 1.69897 / 0.84510 ≈ 2.0103
- Check: 7² = 49, 7³ = 343, so the answer should be just over 2. ✓
5. Useful Special Values
- log_b(1) = 0 for every valid base b, since b⁰ = 1
- log_b(b) = 1 since b¹ = b
- log_b(b^k) = k by the power rule
- b^(log_b(M)) = M by the inverse identity from 2.10
6. Summary
- Product, quotient, and power are the three log properties — all derived from exponent rules
- Logs do not distribute over addition
- Expanding breaks one log into many; condensing combines many into one
- Change-of-base lets you evaluate any logarithm using base 10 or base e
- Memorize log_b(1) = 0 and log_b(b) = 1 — they appear constantly