Example

(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

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

(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.