Answer
$k(\theta) \propto 1 / r$
Work Step by Step
For a curve in polar coordinates, $f(\theta)=r,$ the curvature is:
\[
\frac{|2\left(\frac{d r}{d \theta}\right)^{2}+ r^{2}-r \frac{d^{2} r}{d \theta^{2}} 1}{\left|r^{2}+\left(\frac{d r}{d \theta}\right)^{2}\right|^{3 / 2}}
\]
Since:
\[
\begin{array}{c}
e^{a \theta}=r(\theta) \\
\frac{d y}{d \theta}=a e^{a \theta}=a r(\theta) \\
\frac{d^{2} r}{d \theta^{2}}=\frac{d}{d \theta}(a r(\theta))=a r^{\prime}(\theta)=a^{2} r(\theta)
\end{array}
\]
So:
\[
\begin{aligned}
&\frac{\left|2(a r)^{2}+r^{2}-r\left(a^{2} r\right)\right|}{\left|(a r)^{2}+r^{2}\right|^{3 / 2}}= k(\theta) \\
&=\frac{\left(2 a^{2}-a^{2}+1\right)r^{2}}{r^{3}\left(a^{2}+1\right)^{3 / 2}} \\
\frac{1}{r} \frac{a^{2}+1}{\left(a^{2}+1\right)^{3 / 2}} =\frac{1}{\sqrt{a^{2}+1}}(1 / r)
\end{aligned}
\]
$k(\theta) \propto 1 / r$
