Let be the multiplicative group of positive real numbers, and let
be the additive group of real numbers.
The logarithm function satisfies
for all
, so it is a group homomorphism. The exponential function
satisfies
for all
, so it too is a homomorphism.
The identities and
show that
and
are inverses of each other. Since
is a homomorphism that has an inverse that is also a homomorphism,
is an isomorphism of groups.
Because is an isomorphism, it 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.
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