Can you explain me why this is also isomorphism?

[matematikuvol]

Homomorphism is defined by $f(x*y)=f(x)\cdot f(y)$. One interesting example of this is logarithm function $log(xy)=\log x+\log y$. Can you explain me why this is also isomorphism?

 

[tiny-tim]

hi matematikuvol!

an isomorphism is a homomorphism that is bijective (one-to-one and onto)

(see http://en.wikipedia.org/wiki/Isomorphism)

 

[matematikuvol]

I know that. But I asked only for logarithm because
$$\log (ab)=\log (a)+\log (b)$$
$$\log ((-a)(-b))=\log (a)+\log (b)$$
Why function $f(x)=e^x$ isn't surjective?

 

[tiny-tim]

it's only an isomorphism if you restrict it to the positive numbers

(see also http://en.wikipedia.org/wiki/Isomorphism#Practical_examples)

 

[matematikuvol]

Ok. Tnx for the answer.