predator · importance: high
Logarithm Rules and Properties
Logarithms turn multiplication into addition, which is why they pop up everywhere from sound (decibels) to earthquakes (Richter) to information theory (entropy). The three rules — product, quotient, power — and change of base cover every algebra and pre-calc logarithm problem.
When the method applies
- • Cannot simplify log(xy) or log(x/y)
- • Confused between ln, log, log₁₀
- • Change of base formula trips you up
- • Exponential equations with no obvious solution
Common mistakes
- • Treating log as a multiplication
- • Forgetting the domain (x > 0)
- • Confusing base e (ln) with base 10 (log)
Step-by-step method
- • log(xy) = log(x) + log(y) — product rule
- • log(x/y) = log(x) − log(y) — quotient rule
- • log(xⁿ) = n · log(x) — power rule
- • Change of base: logₐ(x) = ln(x) / ln(a)
- • log(1) = 0 and log(base) = 1 for any base
- • To solve aˣ = b, take ln of both sides: x = ln(b) / ln(a)
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Build long-term fluency
- • Practice rewriting expressions both ways (expanded ↔ combined)
- • Always check domain: argument of log must be > 0
- • Remember ln (natural log) is the inverse of eˣ
Edge cases & deeper reading
For complex-number logs, branch cuts matter — see complex analysis. For multivariable calc, gradient of log probability is critical for ML.
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