logkxlogkb

Logkxlogkb is a compact notation used to denote the product of two logarithms with the same base k, specifically log_k(x) multiplied by log_k(b). The base k is assumed positive and not equal to 1, and the arguments x and b are positive.

In terms of the natural logarithm, log_k(x) log_k(b) can be written as (ln x)(ln b) / (ln k)^2. This representation makes clear that the value depends on the two arguments and on the square of the natural logarithm of the base. The change of base formula implies that for any valid base a, log_k(x) log_k(b) = [log_a(x) log_a(b)] / [log_a(k)^2], so the quantity is independent of the particular base chosen up to the factor (log_a(k))^2.

Basic properties include symmetry and simple special cases. The expression is symmetric in x and b since multiplication is commutative: log_k(x) log_k(b) = log_k(b) log_k(x). It equals zero when either x = 1 or b = 1, because log_k(1) = 0. If x = b, the value is [log_k(x)]^2, a nonnegative quantity for bases where log_k(x) is real.

Usage and notes. Logkxlogkb is not a standard named function in most mathematical texts; it is typically encountered as a shorthand in algebraic manipulations or derivations where the product of two base-k logarithms arises. In formal writing, it is clearer to write log_k(x) log_k(b) rather than a condensed form. See also logarithm, base change formula, and logarithmic identities.