A Graphical view on 2^x=i

Question

I understand how in this equation "$2^x=i$" x equals this $x=(i(1+4n)π)/(2log(2))$ in an algebraic sence But can I have a graphical perspective on this?

Thank you

Answer 1

Graphing complex functions is often hard because you have to represent four real dimensions (for the real and imaginary parts of the input and output) and we can usually only show two or three at a time. One way is to have two separate complex planes and show how points in one map to the other. So for the exponential map $w = e^z$ we can look at, for example, how the horizontal lines $\mathrm{Im}(z) = \beta$ become rays of the form $\mathrm{Arg}(w) = \beta$, and vertical lines $\mathrm{Re}(z) = \alpha$ become circles of the form $|w| = e^\alpha$. These (along with the mapping of some other curves in $\mathbb{R}^2$) are shown in this video from a course on complex analysis.

Then if you want to solve $e^z = w$ for $z$, you can find the circle and ray that intersect at the required value of $w$, such as in the picture below showing how the circle representing $\mathrm{Re}(z) = 0$ and the ray representing $\mathrm{Im}(z) = \frac{\pi}{2}$ intersect at $w = i$. You would find that the rays for $\mathrm{Im}(z) = \pi(\frac{1}{2} + 2n)$ all coincide with each other, giving you multiple solutions.

I used $w = e^z$ to match with the video and to make the expressions easier, but since $a^z = e^{z \ln a}$ for real values $a$, then you would just have some scaling applied to the radii of the circles and rotations applied to the rays which you can adjust for when looking for the right solution.