Generalized Complex Exponentiation

Question

Is there a way to create a general formula for exponentiating two complex numbers like there is for addition and multiplication? Ex:

$(a + bi) + (c + di) = (a + c) + (b + d)i $

$(a + bi) * (c + di) = (a * c - b * d) + (a * d + b * c)i$

$(a + bi) ^ {(c + di)} = ?$

Answer 1

$(a+bi)^{(c+di)}=e^{(c+di)\ln(a+bi)}$

and

$\ln(a+bi)=\ln\left | a+bi \right |+i\arg(a+bi)$

and

$e^{m+ni}=e^m(\cos(n)+i\sin(n)).$