Lognxy

Lognxy, read as log base n of the product x times y, is a mathematical notation for the logarithm of the product xy with base n. It is defined for positive base n with n > 0 and n ≠ 1, and for positive arguments x > 0 and y > 0. The quantity log_n(xy) answers the question: to what exponent must n be raised to produce the product xy?

Key properties: log_n(xy) = log_n x + log_n y; log_n(x^k) = k log_n x; log_n(x/y) = log_n x − log_n y. These follow from the laws of logarithms. The domain requires x > 0 and y > 0; n > 0, n ≠ 1.

Change of base: log_n m = log_k m / log_k n for any base k > 0, k ≠ 1. This allows computation with common bases such as natural logarithm ln or common logarithm log.

Examples: With n = 2, x = 2, y = 8, log_2(xy) = log_2(16) = 4, and equals log_2 2 + log_2 8 = 1 + 3 = 4. Another example: log_10(xy) = log_10 x + log_10 y.

Applications: The concept helps simplify multiplicative relationships, solve exponential equations, and analyze data on logarithmic scales in science and engineering. It also appears in information theory and computer science in the context of growth rates and complexity.