Properties of Logarithms
Turning multiplication into addition, and powers into coefficients.
The explanation
Three properties do all the work, and each mirrors an exponent rule.
Product: log_b(xy) = log_b x + log_b y
Quotient: log_b(x/y) = log_b x − log_b y
Power: log_b(xⁿ) = n·log_b x
The power rule is the useful one for solving equations: it pulls an exponent down where you can reach it with ordinary algebra.
Two things that look like properties but are false:
log(x + y) is NOT log x + log y.
log(x)/log(y) is NOT log(x/y).
Change of base is worth knowing for calculators: log_b x = ln x / ln b.
The properties follow directly from the exponent laws through the inverse relationship. Writing x = b^m and y = b^n, the product xy = b^{m+n} gives log_b(xy) = m + n = log_b x + log_b y. The quotient and power rules follow identically.
Expanding and condensing are the two directions used in practice: expansion separates a complicated argument into a sum for differentiation or estimation, while condensation collapses a sum into a single logarithm so that both sides of an equation can be exponentiated.
The invalid "properties" are instances of the general fact that nonlinear functions do not distribute over addition, the same principle as (a+b)² ≠ a² + b² and √(a+b) ≠ √a + √b.
Applying the power rule requires attention to domain: log(x²) = 2log(x) holds only for x > 0, since the left side is defined for all nonzero x while the right is not. The fully correct identity is log(x²) = 2log|x|.
Worked example
Expand log(x³√y / z²), then condense 2log a − log b.
- Quotient rule: log(x³√y) − log(z²).
- Product and power rules: 3log x + (1/2)log y − 2log z.
- Condensing: 2log a = log(a²).
- Difference becomes a quotient: log(a²/b).
Answer: 3log x + ½log y − 2log z, and log(a²/b)
Common mistakes
- Writing log(x + y) as log x + log y.
- Applying the power rule to a coefficient that multiplies the log rather than the argument's exponent.