log39

Log39 is most commonly used to denote the logarithm with base 39, written as log39(x) or log_39(x). It is the inverse function of the exponential function 39^x and is defined for all positive x. In terms of natural or common logarithms, log39(x) can be expressed by the change-of-base formula: log39(x) = ln(x) / ln(39) = log10(x) / log10(39).

Domain and range: the function is defined for x > 0, and its range is all real numbers. It is strictly increasing because the base 39 is greater than 1. The derivative is d/dx log39(x) = 1 / (x ln(39)).

Key properties include: log39(1) = 0, log39(39) = 1, and log39(a·b) = log39(a) + log39(b); similarly, log39(a^k) = k·log39(a). The inverse function is 39^y, so log39(39^y) = y and 39^(log39(x)) = x.

Examples: log39(39) = 1, and log39(3) ≈ ln(3)/ln(39) ≈ 0.30. As x approaches 0 from the right, log39(x) tends to −∞; as x grows large, log39(x) grows without bound, tracing a smooth increasing curve.

Uses and notes: base-39 logarithms are not common in standard curricula but are mathematically valid and transferable via the change-of-base formula. They can appear in contexts requiring a custom logarithmic scale or in theoretical discussions. Besides the mathematical meaning, the term log39 may occasionally be used as a project or product name in non-mathematical domains, but there is no widely recognized meaning beyond the base-39 logarithm.