Simplifying a complex number into $a + bi$ form I am trying to write $3^i$ in $a + bi $ form. I know 

$$ 3^i = e^{i \log 3 } = e^{ i ( \log 3 + 2 \pi k ) }= e^{i \log 3} e^{2 \pi ki} = e^{i \log 3} = \cos ( \log 3 ) + i \sin( \log 3) $$

is this correct?
Assuming you want all possible values,
$$3^i=e^{i\log3}=e^{i(\ln3+2\pi k i)}$$
and so on (I am using $\ln$ to denote the natural logarithm of a real number).  Note that you have a couple of small errors in your working.
Assuming you want all possible values,
$$3^i=e^{i\log3}=e^{i(\ln3+2\pi k i)}$$
and so on (I am using $\ln$ to denote the natural logarithm of a real number).  Note that you have a couple of small errors in your working.