Why e raise to i tetha is not legal

Today I was solving the equations of quantum mechanics from the book written by Sir David J. Griffith And the old question lingered around my mind Why do we need to use? $$ e^{i\theta} = \cos\theta + i\sin\theta $$ because e (exponential) is a value and angle raise to e makes no sense. Even if it does then \[ \begin {align*} & i\theta = \ln ({\cos\theta + i\sin\theta}) \\ \\ & \theta = \frac{\ln( {\cos\theta + i\sin\theta})}{i} \\ \\ & {\theta}^2 =-(\ln ({\cos\theta + i\sin\theta}))^2 \\ \\ \text {But we know that} \\ \\ & {\theta}^2 \neq -(\ln ({\cos\theta + i\sin\theta}))^2 \\ \\ \end {align*} \]