The value of $a^i$.

Question

I recently tried to compute the value of $a^i$ and $a^{1/i}$ and found $a^i = \cos(\ln(a)) + i\sin(\ln(a))$ as \begin{align} a^i &= e^{\ln(a^i)} = e^{i\ln(a)} \end{align} and using Euler's Formula \begin{align} a^i = e^{i\ln(a)} = \cos(\ln(a))+ i\sin(\ln(a)). \end{align} Does this seem correct?