Logarithms

Idea

Classically, a logarithm is a partially-defined smooth homomorphism from a multiplicative group of number?s to an additive group of numbers. As such, it is a partial section? of an exponential map. As exponential maps can be generalised to Lie groups, so can logarithms.

Definitions

Logarithms of real numbers

Consider the field of real numbers; these numbers form a Lie group under addition (which we will call simply $ℝ$), while the nonzero numbers form a Lie group under multiplication (which we will call ${ℝ}^{*}$). The multiplicative group has two connected components; we will focus attention on the identity component (which we will call ${ℝ}^{+}$), consisting of the positive numbers.

The Lie groups $ℝ$ and ${ℝ}^{+}$ are in fact isomorphic. In fact, there is one isomorphism for each positive real number $b$ other than $1$; this number $b$ is called the base. Fixing a base, the map from ${ℝ}^{+}$ to $ℝ$ is called the real logarithm with base $b$, written $x\mapsto {\log }_{b}x$; the map from $ℝ$ to ${ℝ}^{+}$ is the real exponential map with base $b$, written $x\mapsto b^x$.

The real logarithms are handily defined using the Riemann integral as follows:

Note that $\ln$ is itself a logarithm, the natural logarithm, whose base is $\mathrm{e}=2.71828182845\dots$. (The exponential map may similarly be defined as an infinite series, but I'll leave that for its own article.)

Logarithms of complex numbers

Now consider the field of complex numbers; these also form a Lie group under addition (which we call $ℂ$), while the nonzero numbers form a Lie group under multiplication (which we call ${ℂ}^{*}$). Now the multiplicative group is connected, so we would like to use all of it.

However, $ℂ$ and ${ℂ}^{*}$ are not isomorphic. In fact, not only is there no isomorphism from ${ℂ}^{*}$ to $ℂ$; the only such homomorphism is the zero morphism $x\mapsto 0$. (Essentially, this is because $ℂ$ is simply connected while ${ℂ}^{*}$ is not.) On the other hand, we still have plenty of homomorphisms from $ℂ$ to ${ℂ}^{*}$, one for each nonzero complex number $b$, and these homomorphisms are surjections whenever $b\ne 1$.

So we have these surjections (the complex exponential map with base $b$, for $b\ne 1$), which are regular epimorphisms but not split epimorphisms. However, while they have no sections (being not split), they have quite a few partial section?s, and the domains of the maximal? partial sections are precisely the connected simply connected dense subspaces $R$ of ${ℂ}^{*}$. A complex logarithm with base $b$ on $R$ is this $R$-defined section of the complex exponential map with base $b$.

If $1\in R$, then a complex natural logarithm on $R$ may be defined using the contour integral with the same formula (1) as for the real natural logarithm. We merely insist that the integral be done along a contour within the region $R$. (Since $R$ is connected, there is such a contour; since $R$ is simply connected and $x\mapsto 1/x$ is holomorphic, the result is unique.) Note that if $x\in {ℝ}^{+}\subseteq R$, then the real and complex natural logarithms of $x$ will be equal.

The natural exponential map is periodic? (with period $2\pi \mathrm{i}$), and it is possible to add any multiple of this period to the natural logarithm of any $x\ne 1$ by suitably changing the region $R$. We then obtain the most general notion of maximally-defined complex logarithm with any base by using the formulas

where for $\ln a$ and $\ln b$ we use any previously defined natural complex logarithm.

Logarithms and Lie groups

In the classical examples, the multiplicative groups ${ℝ}^{+}$ and ${ℂ}^{*}$ are both Lie groups. The additive groups $ℝ$ and $ℂ$ are also Lie groups, but they are more than this: they are Lie algebras. (The additive group of a Lie algebra is always a Lie group. Actually, since these are abelian Lie algebras, their Lie-algebra structure is easy to miss, but of course they are vector spaces.) And what's more, each additive group is the Lie algebra of the corresponding Lie group.

This generalises. Given any Lie group $G$, let $𝔤$ be its Lie algebra. Then we have an exponential map $\exp :𝔤\to G$, which is surjective under certain conditions (most famously when $G$ is connected and compact, but also in the classical cases, even though $G$ is not compact). More generally, given any automorphism $\varphi$ of $𝔤$, we have a map $x\mapsto \exp (\varphi (g))$, which is a homomorphism of Lie groups. Any partial section? of this map may be called a logarithm base $\varphi$ on $G$; any partial section of $\exp$ itself may be called a natural logarithm on $G$.