logX

Log base x, written log_x(y), denotes the logarithm of a positive real number y with respect to the base x. It is the inverse function of the exponential with that base: if y = x^t, then t = log_x(y). The notation log_x(y) emphasizes that the base is x and the argument is y.

Domain and base conditions: The base x must be a positive real number different from 1 (x > 0, x ≠ 1). The argument y must be positive (y > 0). The range of log_x is all real numbers.

Key identities: log_x(y) = ln(y) / ln(x), where ln denotes the natural logarithm. This leads to the change-of-base formula: log_x(y) = log_a(y) / log_a(x) for any base a > 0, a ≠ 1. Special values include log_x(1) = 0 and log_x(x) = 1. The expression log_x(y^k) = k log_x(y) and log_x(x^k) = k hold for any real k.

Monotonicity and graphs: If x > 1, the function y ↦ log_x(y) is increasing on its domain. If 0 < x < 1, it is decreasing. As y → 0^+, log_x(y) → -∞ (for x > 1) or +∞ (for 0 < x < 1), and as y → ∞, log_x(y) → ∞ (for x > 1) or −∞ (for 0 < x < 1). The graph is defined only for y > 0 and reflects the inverse of exponential growth or decay with base x.

Relation to other logs: The common logarithm (base 10) is log_10, and the natural logarithm is log_e, often denoted ln. All bases connect via the change-of-base formula.