I have a question about Landau notation for complex analysis.
Let $f(z)$ be a complex function and $g(z)$ be a real valued function of complex valuable which satisfies $g(z) \ge 0$. Then we write $$f(z)=O(g(z))$$ when these functions satisfy $$|f(z)|\le Cg(z)$$ for some constant $C>0$. For example, We use this notation so that \begin{equation} f(z)=g(z)+O(h(z)) \quad (1) \\ e^{f(z)}=e^{g(z)}e^{O(h(z))} \quad (2) \end{equation} for appropriate functions. I think that the second example (2) means $$ |e^{f(z)-g(z)}|\le e^{Ch(z)}.$$
My question is about taking exponential or log of Landau notation. When (1) is true, then we get $$ e^{f(z)}=e^{g(z)}e^{O(h(z))}$$ because $$|e^{f(z)-g(z)}|=e^{\Re(f(z)-g(z))}\le e^{|f(z)-g(z)|} \le e^{Ch(z)}$$ for certain constant $C>0$. But when (2) is true, (1) is not always true. We can get only $$ \log|e^{f(z)-g(z)}| = \Re(f(z)-g(z)) \le \log(ch(z)).$$ We cannot get (1) from (2).
So I think that we can take exponential of Landau notation but we cannot take log. Is this correct? Or Do I have a mistake of my interpretation of the formula (2)?
