Logbxy

Logbxy denotes the logarithm of the product xy with base b, written as log_b(xy). The base b must be a positive real number different from 1, and the argument xy must be positive for real-valued logarithms.

Notation and conventions: The compact form logbxy is a shorthand for log_b(xy). In other contexts the same expression may appear as log_b(xy). Some calculators or programming languages use log(b, xy) to indicate a base-b logarithm.

Domain and basic limits: For real-valued logarithms, the base satisfies b > 0 and b ≠ 1, and the argument xy must be greater than 0. This means xy > 0, so x and y must have the same sign (both positive or both negative) if one is to consider the product argument in the real numbers.

Key property and caveats: If x > 0 and y > 0, then log_b(xy) = log_b x + log_b y. This follows from the exponent rules. If either x or y is not positive, log_b x or log_b y is not defined in the real numbers, so the additive identity does not apply in the real setting; however log_b(xy) remains defined whenever xy > 0.

Computation: log_b(xy) can be computed as log_b(xy) = ln(xy) / ln(b) = (ln x + ln y) / ln(b), provided x > 0 and y > 0. Using change of base with natural logarithms is a common approach in practice.

Examples: log_2(9) ≈ 3.16993, since 9 = 3 × 3 and log_2(9) = log_2 3 + log_2 3. More generally, if x and y are positive, log_b(xy) equals the sum of their separate logs base b. See also log, log_b, and the product rule for logarithms.