  Many physical systems share the property of scale invariance. Most of them
show ordinary power-law scaling, where quantities can be expressed as a leading
power law times a scaling function which depends on scaling-invariant ratios of
the parameters. However, some systems do not obey power-law scaling, instead
there is numerical evidence for a logarithmic scaling form, in which the
scaling function depends on ratios of the logarithms of the parameters. Based
on previous ideas by C. Tang we propose that this type of logarithmic scaling
can be explained by a concept of local scaling invariance with continuously
varying exponents. The functional dependence of the exponents is constrained by
a homomorphism, which can be expressed as a set of partial differential
equations. Solving these equations we obtain logarithmic scaling as a special
case. The other solutions lead to scaling forms where logarithmic and power-law
scaling are mixed.
