Kantalogaritmi

Kantalogaritmi, in mathematics, refers to the logarithm with a specified base. It is the inverse function of the exponential function b^x, where b is the base, a positive real number different from 1. The kantalogaritmi log_b(x) is defined for x > 0 and yields real values. The base b determines the function’s growth and monotonicity: it is increasing if b > 1 and decreasing if 0 < b < 1.

Key properties include:

- log_b(1) = 0 and log_b(b) = 1.

- log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x) for x > 0.

- Change of base: log_b(x) = ln(x) / ln(b) = log_k(x) / log_k(b) for any base k > 0, k ≠ 1.

- Domain and range: the domain is x > 0, and the range is all real numbers.

Base considerations and representations:

Common choices are b = 10 (common logarithm) and b = e (natural logarithm, denoted ln). The concept of kantalogaritmi encompasses any base b > 0, b ≠ 1, and is used to solve exponential equations, model multiplicative growth, and transform multiplicative relationships into additive ones.

Examples:

log_2(8) = 3, log_10(100) = 2, log_{1/2}(4) = -2.

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