Curves and vectorial functions

For any negative number $a$, the following parametric equations define a logarithmic spiral:

$$x = e^{at} \cos(t) \qquad y = e^{at} \sin(t) \qquad 0 \leq t < \infty$$A particular spiral is:

Notice that $t$ lies in the unbounded interval $[0, \infty)$. A surprising fact about this curve is that it has a finite length!

For now, note that the distance from any point $(x, y)$ to the origin is given as $x^2 + y^2$.

For any time $t$, input the logarithm of the distance from the position of the curve at time $t$ to the origin.