Editing Isomorphism (section)

===Logarithm and exponential===
Let <math> \R ^+ </math> be the [[multiplicative group]] of [[positive real numbers]], and let <math>\R</math> be the additive group of real numbers.

The [[logarithm function]] <math>\log : \R^+ \to \R</math> satisfies <math>\log(xy) = \log x + \log y</math> for all <math>x, y \in \R^+,</math> so it is a [[group homomorphism]]. The [[exponential function]] <math>\exp : \R \to \R^+</math> satisfies <math>\exp(x+y) = (\exp x)(\exp y)</math> for all <math>x, y \in \R,</math> so it too is a homomorphism.

The identities <math>\log \exp x = x</math> and <math>\exp \log y = y</math> show that <math>\log</math> and <math>\exp </math> are [[inverse function|inverses]] of each other. Since <math>\log</math> is a homomorphism that has an inverse that is also a homomorphism, <math>\log</math> is an [[Group isomorphism|isomorphism of groups]], i.e., <math>\R^+ \cong \R</math> via the isomorphism <math>\log x</math>.

The <math>\log</math> function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a [[ruler]] and a [[table of logarithms]], or using a [[slide rule]] with a logarithmic scale.