The function \(\exp:(\mathbb{R},+)\to(\mathbb{R}^*,\cdot)\), \(x\mapsto e^x\), is an injective homomorphism with image \(\mathbb{R}_{>0}\). Which group is isomorphic to \((\mathbb{R},+)\)?
The kernel is \(\{0\}\), since \(e^x=1\) only if \(x=0\). Hence \(\mathbb{R}\cong\mathbb{R}/\{0\}\cong\operatorname{im}\exp=\mathbb{R}_{>0}\). It is not \(\mathbb{R}^*\) because the exponential never takes negative values.
Every group homomorphism \(\varphi:G\to H\) induces an isomorphism between the quotient of \(G\) by the kernel of \(\varphi\) and the image of \(\varphi\).