(a) Graph the equation \(r=e^{\theta /5}\).

(b) Find parametric equations for \(r=e^{\theta /5}\).

Graphs of polar equations of the form \(r=e^{\theta /a},\) \(a>0,\) are called logarithmic spirals, since the equation can be written as \(\theta =a \ln r\). A logarithmic spiral spirals infinitely both toward the pole and away from it.

Solution The polar equation \(r=e^{\theta /5}\) lacks the symmetry you may have observed in the previous examples. Since there is no number \(\theta\) for which \(r=0\), the graph does not contain the pole. Also observe that:

\(r\) is positive for all \(\theta\).

\(r\) increases as \(\theta\) increases.

\(r\rightarrow 0\) as \(\theta \rightarrow -\infty\).

\(r\rightarrow \infty\) as \(\theta \rightarrow \infty \).

We use a calculator to obtain Table 6. Figure 47(a) shows part of the graph \(r=e^{\theta /5}.\) Figure 47(b) shows the graph for \(\theta =-\dfrac{3\pi}{2}\) to \(\theta =2\pi\) using technology.

TABLE 6

\(\theta\)	\(r=e^{\theta /5}\)	\(( r,\theta )\)
\(-\dfrac{3\pi }{2}\)	\(\ 0.39\)	\(\left( 0.39,-\dfrac{3\pi }{2}\right)\)
\(-\pi\)	\(\ 0.53\)	\(( 0.53,-\pi )\)
\(-\dfrac{\pi }{2}\)	\(\ 0.73\)	\(\left( 0.73,-\dfrac{\pi }{2}\right)\)
\(-\dfrac{\pi }{4}\)	\(\ 0.85\)	\(\left( 0.85,-\dfrac{\pi }{4}\right)\)
\(0\)	\(\ 1\)	\(\left( 1,0\right)\)
\(\dfrac{\pi }{4}\)	\(\ 1.17\)	\(\left( 1.17,\dfrac{\pi }{4}\right)\)
\(\dfrac{\pi }{2}\)	\(\ 1.37\)	\(\left( 1.37,\dfrac{\pi }{2}\right)\)
\(\pi\)	\(\ 1.87\)	\(( 1.87, \ \pi )\)
\(\dfrac{3\pi }{2}\)	\(\ 2.57\)	\(\left( 2.57,\dfrac{3\pi }{2}\right)\)
\(2\pi\)	\(\ 3.51\)	\(( 3.51, \ 2\pi )\)

The spiral \(r=e^{\theta /5}\).

(b) We obtain parametric equations for \(r=e^{\theta /5}\) by using the conversion formulas \(x=r \ \cos \theta\) and \(y=r \ \sin \theta\): \[ x=r \ \cos \theta =e^{\theta /5} \ \cos \theta \qquad y=r \ \sin \theta =e^{\theta /5}\sin \theta \]

where \(\theta\) is the parameter and \(\theta\) is any real number.