Here is the graph you want ($r=e^{\theta}$):
Here is the graph you get from your attempted solution ($4^{\tan^{-1}(y/x)}=x^2+y^2$):
Your equation has three problems. First, you get two spirals instead of the one you desire. Second, you get only part of the spiral, because $\theta$ is too limited. Third, if the spiral continued, you would get holes at any points where $x=0$.
These problems have the same cause: using the standard arctangent function of $y/x$. The arctangent function does not distinguish between points in the first and third quadrants, or between the second and fourth quadrants. This gives you the two spirals. (This is what @Narasimham is referring to in his answer.) Even if you ignore this,
$\tan^{-1}\frac yx$ is not quite equal to $\theta$. The arctangent function is indeed a function, so it limits theta to $-\pi/2<\theta<\pi/2$. The atan2 function expands the range to $-\pi<\theta\le\pi$. But polar graphing is not limited to either of those ranges for theta. Last, using the standard arctangent function requires you to use $y/x$, which is not defined for $x=0$, since there is a division by $x$.
We can remove the first and third problems by using $\mathrm{atan2(x,y)}$ rather than $\tan^{-1}\frac yx$. The second problem is removed by looking at the remainder of the angles after division by $2\pi$. Unfortunately, the atan2 function gives the wrong range to do this conveniently, so we must check its remainder as well.
Here is my Cartesian equation for your graph.
$$\mathrm{fract}\left(\frac{\mathrm{atan2}(x,y)-\log_4(x^2+y^2)}{2\pi}\right)=0$$
or perhaps
$$\mathrm{mod}(\mathrm{atan2}(x,y)-\log_4(x^2+y^2),2\pi)=0$$
Unfortunately, I do not have a graphing program that both graphs general Cartesian relations and allows the atan2 function. The best I can do replaces $\mathrm{atan2}(x,y)$ with $\mathrm{if}(x>0,\tan^{-1}(y/x),\tan^{-1}(y/x)+\pi)$, which leaves some artefacts in my grapher.
Could someone graph this for me and confirm that it is correct? Also, be careful in the use of atan2. Some environments use $\mathrm{atan2}(x,y)$ while others use $\mathrm{atan2}(y,x)$. Make sure your parameters are in the correct order for your grapher.