Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition. Does there exists an Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.
Attempt
In my opinion It does not exists.
If there exists such an isomorphism say $\phi :\mathbb{C}^* \to \mathbb{C}$ then, from the fact that Isomorphism takes identity to identity we have $$\phi \left( 1 \right)=0$$
From the fact 

Given $\phi:G \to H$ be an isomorphism, then For a fixed integer $k$ and a fixed group element $b$ in $G$, the
  equation $x^k=b$ has the same number of solutions in $G$ as does
  the equation $x^k=\phi\left(b\right)$ in $H$.

We have $x^4=1$ and $x^4=0$ must have same number of solutions, but the solutions to these equations are 4 and 1 only.   
 A: No, no such isomorphism exists.
For if
$\phi: \Bbb C^\ast \to \Bbb C \tag 1$
were such an isomorphism, it would, as pointed out by our OP Rakesh Bhatt, satisfy
$\phi(1) = 0; \tag 2$
one may argue this in the usual way, via the fact that group isomorphisms take identities to identities, or even more generally by observing that
$\phi(1) = \phi(1^2) = \phi(1) + \phi(1), \tag 3$
whence
$\phi(1) = 0. \tag 4$
In any event,we may use (2), (4) to resolve our problem as follows: let $1 
\ne \omega \in \Bbb C^\ast$ be an $n$-th root of unity where $n > 1$;. then
$\omega^n = 1, \tag 5$
whence
$\phi(\omega^n) = \phi(1) = 0; \tag 6$
but
$\phi(\omega^n) = \phi(\omega) + \phi(\omega) + \ldots + \phi(\omega), \; n \; \text{times} = n \phi(\omega); \tag 7$
it follows from (6), (7) that 
$n \phi(\omega) = 0, \tag 8$
whence
$\phi(\omega) = 0; \tag 9$
since $\phi$ is an isomorphism, we now conclude from (2) and (9) that
$\omega = 1, \tag{10}$
contrary to our choice of $\omega$; thus no such isomorphism $\phi$ can exist.
