log1010n

Log1010n is a notation used to denote the logarithm of a positive number n with base 1010. In this form, the base is 1010, which is greater than 1, so the function is increasing with respect to n and has the usual logarithmic growth behavior.

Definition and basic properties: log1010(n) can be defined as log1010(n) = ln(n) / ln(1010), where ln denotes the natural logarithm. The domain is n > 0, and the range is the set of all real numbers. Key values include log1010(1) = 0 and log1010(1010) = 1. Because the base exceeds 1, the function is monotone increasing. Exponentiation provides the inverse relation: 1010^{log1010(n)} = n. The change of base formula gives log1010(n) = logb(n) / logb(1010) for any positive base b ≠ 1, and a common specialization is log1010(n) = log10(n) / log10(1010).

Ambiguity and notation: notation like log1010n can be ambiguous without parentheses. In some contexts it might be read as a log of 1010n with a different base, so precise parentheses are important. When clarity is required, standard forms such as log base 1010 of n (log1010(n)) or log base b of n (logb(n)) are preferred.

Applications: This base is often used in discussions of slow-growing logarithmic scales, algorithmic complexity comparisons, and change-of-base calculations. In practice, log1010(n) grows more slowly than common logarithms such as log10(n) because the base 1010 is large.