It is standard practice to define on $\mathbb{C}$, $\operatorname{Log}(z) = \log(|z|) + i \operatorname{Arg}(z).$ When composed with $\exp$, we get $\operatorname{Log} \circ \exp (z) = z$, the identity function, for all $z$ in the $2\pi $-wide strip $\{ z\, :\, 0 < \Im(z) < 2\pi \}$.
Now, on the one hand, in case two analytic functions are identical on an open set, then they are identical. On the other hand, $\operatorname{Log} \circ \exp$ is certainly not the identity function throughout its domain. Where is the faulty deduction ?