Can you explain me why this is also isomorphism?

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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?
 
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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?
 


Ok. Tnx for the answer.
 

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