Sketch the families of level curves of $u$ and $v$ for the function $f=u+iv$ given by $f(z)=e^{\alpha z}$, where $\alpha$ is complex.
My work so far: \begin{align*} e^{\alpha z} &= e^{(a+ib)(x+iy)} \quad (a,b \in \mathbb{Z}) \\ &= e^{ax-by} e^{i(ay+bx)} \\ &= e^{ax-by} \cos(ay+bx) + i e^{ax-by} \sin(ay+bx) \end{align*}
Since $f$ is a complex mapping, we can write $f(z)=u(z)+iv(z)$, namely $$u(x,y)= e^{ax-by} \cos(ay+bx), \qquad v(x,y)= e^{ax-by} \sin(ay+bx)$$ Now, how can I sketch level curves? I set $u=1$ for example, and I can't solve for $y$ in terms of $x$. I even had trouble doing this on the Desmos graphing calculator.