logkxy

Log_k(xy) denotes the logarithm of the product of x and y with base k, written as log_k(xy). In real-valued arithmetic, this expression is defined when the base k is a positive real number not equal to 1 (k > 0 and k ≠ 1) and the argument xy is positive (so x > 0 and y > 0).

A fundamental property is the product rule: log_k(xy) = log_k x + log_k y for x > 0 and y > 0. This follows from the general properties of logarithms and is valid whenever the individual logarithms log_k x and log_k y are defined. Through a change of base, log_k(xy) can be written as ln(xy)/ln(k) = (ln x + ln y)/ln(k).

As a function of x and y, with a fixed base k, the domain is x > 0 and y > 0, and the range is all real numbers, since ln x and ln y can be adjusted to yield any desired sum. If k > 1, log_k is increasing; if 0 < k < 1, it is decreasing.

Log_k relates to the natural logarithm by the identity log_k t = ln t / ln k for any t > 0. This makes log_k(xy) convenient for converting products into sums, mirroring the more general log rules.

Applications of log_k(xy) appear in algebra and analysis, where simplifying expressions involving products, multiplicative processes, or likelihoods in statistics can benefit from converting a product into a sum via the logarithm.