Show that the mapping a log10a is an isomorphism from R unde

Show that the mapping a log₁₀a is an isomorphism from R⁺ under multiplication to R under addition.

R is the set of real numbers obviously.

Solution

Let’s call the map in question . Now is obviously well defined (there is no ambiguity about its definition), and it is not difficult to see that it is a homomorphism if we remember a little fact from calculus 142 (that log10(ab) = log10(a) + log10(b) for all a, b R +). So for a, b R + we have (ab) = log10(ab) = log10(a) + log10(b) = (a) + (b) as required. Now if a R then 10a R+ and (10a ) = log10(10a ) = a, so is onto, and if log10(a) = log10(b), then e log10(a) = e log10(b) , that is a = b, so the map is injective. We conclude that is an isomorphism.