log10aH

In mathematics and applied sciences, log10(aH) denotes the base-10 logarithm of the product aH. Here a and H are positive real numbers (or dimensionless quantities). By definition, log10(aH) = log10(a) + log10(H).

Because the logarithm requires a dimensionless argument, if a and H carry units, the product must be nondimensionalized before evaluating the log. One common approach is to use a dimensionless form log10((a/a0)(H/H0)) where a0 and H0 are reference values with the same units.

Properties and domain: log10(aH) is defined for aH > 0 and is a monotonically increasing function of its argument. Real-valued logs require aH to be positive; if either a or H is zero or negative, the real logarithm is undefined. Complex logarithms can be used when the argument is nonpositive, but they involve branch cuts and imaginary components.

Applications and interpretation: log10(aH) is often encountered in multiplicative models where a and H are factors that influence a quantity on a logarithmic, additive scale. Taking the logarithm converts multiplicative relationships into additive ones, facilitating linear modeling, data transformation, and comparison on a compressed scale. For example, if a = 2 and H = 100, then aH = 200 and log10(aH) ≈ 2.3010, which equals log10(a) + log10(H) = log10(2) + log10(100).

See also: base-10 logarithm, logarithm properties, dimensional analysis.