Log10log102

Log10log102 denotes the nested base-10 logarithm of 2, written as log10(log10(2)). It is the composition of the base-10 logarithm applied first to 2 and then again to the result. In standard mathematical notation, this expression is log10(log10(2)).

Evaluating the expression starts with the inner logarithm: log10(2) ≈ 0.30103. Since this inner value is between 0 and 1, its base-10 logarithm is negative. Therefore, log10(log10(2)) ≈ log10(0.30103) ≈ -0.52138. The final value is a negative real number, reflecting the fact that the inner result lies in the interval (0, 1).

Domain and general considerations: The outer logarithm requires its input to be positive. For log10(log10(a)) to be defined, one needs log10(a) > 0, which occurs when a > 1. In general, the function f(a) = log10(log10(a)) is increasing on (1, ∞); it is negative for 1 < a < 10, zero at a = 10, and positive for a > 10. For the specific case a = 2, the value is negative as noted.

Context and usage: Nested logarithms like log10(log10(2)) arise in mathematical analysis and data scaling where quantities span multiple orders of magnitude. They can help in compressing wide ranges and in studying asymptotic behavior, though for concrete calculations the inner and outer logarithms are typically computed with standard numerical methods or calculators.