Log15x

Log15x, or log base 15 of x, is the logarithmic function denoted log_15(x). It is defined for x > 0 and is the inverse of the exponential function 15^t. Key reference values include log_15(15) = 1 and log_15(1) = 0.

The base 15 being greater than 1 makes the function increasing. As x grows, log_15 x increases slowly, reflecting the growth rate of logarithms. The domain is x > 0 and the range is all real numbers.

Properties and rules follow the standard logarithm identities: log_15(xy) = log_15 x + log_15 y; log_15(x^k) = k log_15 x; log_15(x/y) = log_15 x − log_15 y. Change of base allows conversion to any other base: log_15 x = ln x / ln 15 = log_k x / log_k 15 for any base k > 0, k ≠ 1.

Calculus and graphical notes: the derivative is d/dx log_15 x = 1/(x ln 15), which is positive for x > 0. The second derivative is −1/(x^2 ln 15), indicating concavity downward on the positive real axis. The graph passes through (1, 0) and grows without bound, albeit slowly.

Usage and context: log_15 x is used wherever a base-15 scale or a problem structure with a base-15 relationship arises. While 15 is not a common base in standard applications, the logarithm with any base is a valid tool for solving exponential equations and for transforming multiplicative relationships into additive ones.