13.5
Consider the logarithm function $log=exp^{-1}$ on $\mathbb{R}$. Prove:
a) There are of definition of log is $\mathbb{R}_{+} :=]0,\infty]$

Question

13.5
Consider the logarithm function $log=exp^{-1}$ on $\mathbb{R}$. Prove:
a) There are of definition of log is $\mathbb{R}_{+} :=]0,\infty]$

anonymous · Answer

@satellite73 welcome, long time did not seen u :)

anonymous · Answer

hello

anonymous · Answer

hello

anonymous · Answer

i am not sure what your definition of $\exp$ is my but my guess is that somewhere in the definition your know that 
a) $\exp$ is injective (one to one) so inverse exists and 
b) the range of $\exp$ is $]0,\infty)$ making that the domain of $\exp^{-1}$

anonymous · Answer

@satellite73  pls can you show it mathematically, so that i can give to my mentor, this is my last homework and i need some points for be able to participating in exam.

anonymous · Answer

i don't know how to show it mathematically other than to say what i wrote above. 
in general if $f:A\to B$ is injective, then $f^{-1}$ exists and the domain of $f^{-1}$ is the codomain of $f$

anonymous · Answer

ahh you mean thats enough and its solution ?

anonymous · Answer

you should say that for all $y\in ]0,\infty)$ exists $x\in \mathbb{R}$ such that $\exp(x)=y$ and therefore by definition of the inverse $\log(y)=x$
 but this is a very general fact. if $f$ has and inverse $f^{-1}$ it is always the case that the image of $f$ is the domain of $f^{-1}$ and the domain of $f$ is the image of $f^{-1}$

anonymous · Answer

ok satellite thank you very much, its much appreciated

anonymous · Answer

yw