logbasen

logbasen, in mathematics, refers to the logarithm with a specified base b. The logarithm of a positive number x with base b is the exponent y that satisfies b^y = x. It is written as log_b(x). The base b must be a real number greater than 0 and not equal to 1. The logarithm is the inverse function of the exponential function b^x.

The base determines the shape and behavior of the logarithm. If b > 1, log_b is increasing; if 0 < b < 1, log_b is decreasing. The domain of log_b is x > 0, and its range is all real numbers. Important values include log_b(1) = 0 and log_b(b) = 1.

Key properties include the product, quotient, and power rules: log_b(xy) = log_b(x) + log_b(y); log_b(x^k) = k log_b(x); log_b(x/y) = log_b(x) − log_b(y). The change of base formula allows converting logs between bases: log_b(x) = log_k(x) / log_k(b) for any positive base k ≠ 1. Common choices are the natural logarithm, ln(x) = log_e(x), and the common logarithm, log_10(x) = log_10(x).

Examples help illustrate: log_2(8) = 3, since 2^3 = 8; log_10(1000) = 3, since 10^3 = 1000; ln(e^2) = 2, since e^2 = e^2. In real-valued contexts, the logarithm is defined for x > 0; extensions to complex numbers require broader definitions.

Applications of logbasen appear across science and engineering, including modeling exponential growth and decay, data transformation, and analyzing algorithms and scales where multiplicative relationships become additive.