log100y

The expression log100y denotes the logarithm of the variable y taken with respect to base 100. In standard notation this is written as log₁₀₀(y). It is defined for all positive real numbers y and is related to the natural logarithm by the change‑of‑base formula: log₁₀₀(y) = ln(y)/ln(100). Since ln(100) equals 2 ln(10), the expression can also be written as log₁₀₀(y) = log₁₀(y)/2, where log₁₀ is the common logarithm base 10. Thus a log base 100 value is exactly half the common logarithm of the same number.

Logarithms to base 100 are not as common in modern practice as base 10 or base e, but they have appeared in certain historical and engineering contexts. One notable use was in the development of logarithmic tables for computational purposes, where base 100 logarithms allowed for a compact representation of large numbers by reducing the number of digits needed. In this setting, two decimal digits in a base‑10 logarithm correspond to a single digit in a base‑100 logarithm. The property that log₁₀₀(10) = 0.5, log₁₀₀(100) = 1 and log₁₀₀(1,000) = 1.5 demonstrates that powers of ten are neatly scaled by increments of 0.5 in base‑100 logarithms.

Mathematically, log₁₀₀ preserves all the standard properties of logarithms: the product rule, quotient rule, and power rule apply unchanged. For example, log₁₀₀(xy) = log₁₀₀(x) + log₁₀₀(y) and log₁₀₀(xⁿ) = n·log₁₀₀(x). Because of the straightforward scaling relation with log₁₀, any calculation involving a base‑100 logarithm can be translated into a base‑10 calculation by dividing the result by two, which makes the base‑100 form particularly convenient for certain algebraic manipulations.

In current usage, base‑100 logarithms are rarely cited explicitly; instead, practitioners work with standard bases and convert as needed. Nonetheless, the concept remains a useful example of how changing the logarithm base modifies the scale of the output, and it illustrates the general principle of base change in logarithmic functions.