log13

log13 typically refers to the logarithm with base 13, written as log base 13 of x, or log_13(x). It is the exponent y that satisfies 13^y = x for a given positive x. Thus x > 0 and y = log_13(x).

In standard notation, log_13(x) can be computed using any common logarithm via the change-of-base formula: log_13(x) = ln(x) / ln(13) = log10(x) / log10(13). This makes it easy to evaluate with calculators or software that provide natural or common logarithms.

Basic values and properties: log_13(1) = 0 and log_13(13) = 1, since 13^0 = 1 and 13^1 = 13. For powers of 13, log_13(13^k) = k. Concrete approximations include log_13(2) ≈ 0.270, log_13(3) ≈ 0.428, and log_13(10) ≈ 0.898. Because the base 13 is greater than 1, the function log_13(x) is increasing on the positive real numbers, and it satisfies the usual logarithmic rules: log_13(xy) = log_13(x) + log_13(y) and log_13(x^a) = a log_13(x).

Applications and context: log_13 is used to solve exponential equations where the unknown appears as an exponent with base 13, to compare growth rates in models incorporating base-13 scaling, and in educational contexts as an example of a log with a non-standard base. It is related to other logarithms through the change-of-base formula, enabling calculations with base 13 using more common logarithms.