Abstract Algebra: Proving G is Isomorphic to H with Log

[halvizo1031]

1. Homework Statement [/b]
The set of positive real numbers, R+, is a group under normal multiplication. The set of real
numbers, R, is a group under normal addition. For the sake of clarity, we'll call these groups G and H respectively.
Prove that G is isomorphic to H under the isomorphism log. (When I write
"log" it means "log(base 10)10" and you should think of it as a function/mapping from G to H).
You don't need to go overboard about proving one-to-one and onto... just appeal to things
you know about the log function.

Homework Equations

can someone please help me on this?

The Attempt at a Solution

 

[Robert1986]

To prove that a transformation is an isomorphism, you need to prove that it is a homomorphism. If $\phi: G \to H$ is the transformation, this means:

$\phi(ab) = \phi(a) + \phi(b)$

Do you see why this works for the log function? Next, you need to prove that it is 1-1 and onto.

 

[halvizo1031]

I'm sorry, but I still do not see how this applies to logarithms...how do logarithms apply in showing that R+ maps to R?

 

[halvizo1031]

also, how can i identify an isomorphism from H onto G? can i just say phi(a+b)=10^(a+b)=10^a times 10^b = phi(a) times phi(b). therefore, log (a+b) = log(a)log(b)?

 

[HallsofIvy]

It seems to me very strange that you would be doing a problem in abstract algebra and yet NOT know the properties of logarithms.

It is log(ab)= log(a)+ log(b), not "log(a+ b)= log(a)log(b).

Further since log(x) is only defined for positive x, you must have log(x) taking positive real numbers to real numbers, G--> H in your notation. You have your function going the wrong way.