logkx
Logkx, written as log_k(x) or log_k x, denotes the logarithm of x with base k. It is defined for base k > 0 with k ≠ 1 and for x > 0, and it is the inverse of the exponential function k^t. In other words, log_k x equals the exponent to which k must be raised to obtain x. This yields log_k(k) = 1 and log_k(1) = 0. Change of base: log_k x = ln x / ln k = log_a x / log_a k for any positive a ≠ 1. Basic rules include log_k(x^n) = n log_k x, log_k(xy) = log_k x + log_k y, and log_k(1/x) = -log_k x. Monotonicity depends on k: the function is increasing for k > 1 and decreasing for 0 < k < 1. Common special cases include the natural logarithm ln x = log_e x and the common logarithm log_10 x. Examples: log_2 8 = 3, log_10 100 = 2. The graph is defined on x > 0 and tends to −∞ as x → 0+, while it tends to ∞ as x → ∞ for bases greater than 1; for 0 < k < 1 the graph is reversed.
Applications include solving exponential equations, transforming multiplicative relationships into additive ones, and working with logarithmic scales