logT

Logt is not a standard standalone term in mathematics, but in many contexts the expression log_t(x) denotes the logarithm of x with base t. The base t must satisfy t>0 and t≠1. Log_t(x) is defined as the unique real number y such that t^y = x for x>0. This makes log_t(1) = 0 and log_t(t) = 1.

Several useful properties mirror those of other logarithms. For any positive x and any real exponent r, log_t(x^r) = r·log_t(x). The product rule holds: log_t(xy) = log_t(x) + log_t(y). The change of base formula allows conversion to any other base: log_t(x) = ln(x)/ln(t) = log_b(x)/log_b(t) for any base b>0, b≠1. The base also determines monotonicity: if t>1, log_t is increasing; if 0<t<1, it is decreasing.

In practice, log_t notation appears mainly in mathematical writing to emphasize a specific base. Many calculators and programming languages implement logarithms with a fixed base (such as base 10 or the natural base e) and provide mechanisms to compute other bases via division, e.g., log_t(x) = ln(x)/ln(t). When writing code or documentation, it is important to clarify the base to avoid ambiguity, since some contexts use log for base 10, others for base e, and some may require explicit base notation such as log_t or log(x, t).

Overview: log_t is the base-t logarithm, a fundamental inverse of the exponential function t^y, with standard properties and the change-of-base relationship enabling computations across bases.