log21

Log21 denotes the logarithm with base 21, the inverse function of the exponential 21^x. It is usually written as log base 21 of x, written as log_21(x) or, in some contexts, log21(x). The domain is x > 0, and its range is all real numbers. The function is defined for positive inputs and returns real outputs that satisfy 21^y = x.

For bases greater than 1, such as 21, log_21 is increasing and continuous on its domain. Key values include log_21(1) = 0 and log_21(21) = 1. As x approaches 0 from the right, log_21(x) tends toward negative infinity; as x grows without bound, log_21(x) grows without bound as well. The graph is a smooth curve passing through (1,0) and (21,1), with a vertical asymptote at x = 0.

Change of base formula allows computation using common bases such as natural or common logarithms: log_21(x) = ln(x)/ln(21) = log_10(x)/log_10(21). This enables numerical evaluation with standard calculators. For example, log_21(2) ≈ 0.2276 and log_21(441) = 2, since 21^2 = 441.

Log_21 is part of the broader family of logarithms log_b(x) for base b > 0, b ≠ 1. While base 21 is not among the most common choices, the same properties and rules apply, including the product, quotient, and power rules, and the change-of-base formula.