logbN

Logbn denotes the logarithm of n with base b, written as log_b(n). It is the inverse of the exponential function with the same base, b^x. By definition, log_b(n) = x if and only if b^x = n, with n > 0, b > 0, and b ≠ 1.

Domain and range: The argument n must be positive, while the base b must satisfy b > 0 and b ≠ 1. Under these conditions log_b(n) can take any real value. The function is increasing when b > 1 and decreasing when 0 < b < 1.

Change of base and identities: A common tool is the change-of-base formula log_b(n) = ln(n)/ln(b) for any base k, e.g., k = e or k = 10. Other useful identities include log_b(1) = 0, log_b(b) = 1, log_b(xy) = log_b(x) + log_b(y), and log_b(x^k) = k log_b(x).

Examples and applications: log_2(8) = 3 and log_10(100) = 2. Logarithms are used to solve exponential equations, analyze growth and decay, and measure information content with different bases (bits for base 2, natural units for base e, and common units for base 10).