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log(xy) = log(x) + log(y). The log of a product equals the sum of the logs. Works for any base.
log(x/y) = log(x) - log(y). The log of a quotient equals the difference of the logs.
log(x^n) = n * log(x). The log of a power equals the exponent times the log of the base.
log_b(x) = ln(x) / ln(b). Convert between any log bases using natural log or common log.
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Use the three log properties: (1) Product rule: log(xy) = log(x) + log(y), (2) Quotient rule: log(x/y) = log(x) - log(y), (3) Power rule: log(x^n) = n*log(x). Apply them in order to break a single log into multiple terms.
Condensing is the reverse of expanding. Combine log terms: (1) Coefficients become exponents: 2*log(x) = log(x^2), (2) Addition becomes multiplication: log(x) + log(y) = log(xy), (3) Subtraction becomes division: log(x) - log(y) = log(x/y).
log_b(x) = log_c(x) / log_c(b) for any base c. Most commonly, log_b(x) = ln(x) / ln(b) or log_b(x) = log(x) / log(b). This lets you evaluate any logarithm using a calculator.
The natural log (ln) uses base e (approximately 2.718). It follows the same properties as any log: ln(xy) = ln(x) + ln(y), ln(x/y) = ln(x) - ln(y), ln(x^n) = n*ln(x). Additionally, ln(e) = 1 and ln(1) = 0.