logelog10

Logelog10 is not a standard, widely recognized mathematical term. In practice, it is sometimes used to describe a nested logarithmic construct involving logarithms of different bases, most commonly in two interpretations: the natural logarithm of the base-10 logarithm, or the base-10 logarithm of the natural logarithm. Each interpretation has its own domain and algebraic properties.

One interpretation is log_e(log_10(x)), written more compactly as ln(log10(x)). This is the natural logarithm of the base-10 logarithm of x. Its domain requires x > 0 and log10(x) > 0, which means x > 1. A useful identity is ln(log10(x)) = ln(ln(x)) − ln(ln(10)). As x grows, ln(log10(x)) increases slowly; as x approaches 1 from the right, ln(log10(x)) tends to negative infinity.

Another interpretation is log_10(ln(x)), written as base-10 logarithm of the natural logarithm of x. This requires x > 1, since ln(x) > 0 is necessary for the outer logarithm. It can be expressed as log10(ln(x)) = ln(ln(x)) / ln(10). This function also increases slowly with x; as x approaches 1+, ln(x) approaches 0+, and log10(ln(x)) tends to negative infinity.

Because the notation is ambiguous, readers should verify the intended meaning in any given text. When working with nested logs, it is helpful to apply the change-of-base formula to relate different bases and to note the domain restrictions for each nesting. If a source uses logelog10 without clarification, consult the author or references to determine whether it denotes ln(log10(x)) or log10(ln(x)) and proceed accordingly.