(The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.)

We raise both sides to the power of 1 (strictly decreasing function in positive reals):

An important example of a logarithmically convex function is the gamma function on the positive reals (see also the Bohr-Mollerup theorem).

The logarithmic prior on the positive reals.

First, observe that the range of r is all positive reals except for 0.

Quotienting out by the positive reals then yields another copy of the real line, with marked points at the integers.

The Connes spectrum, a closed subgroup of the positive reals, is obtained by applying the exponential to the kernel of this flow.

Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals.

When replacing by its power series in the integral definition of , one obtains (assume "x,s" positive reals for now):

Represent magnitudes (positive reals) as nonempty proper initial segments of the positive rationals with no largest element.