A question about the polar equation $r=ln(\theta)$

Question

Take the polar equation r=ln(theta) for theta between $0$ and $2\pi$. 1 In this, we have a loop that connects at a point $(x,y)$ for $2$ unique points $(r_1,\theta_1)$ and ($r_2,\theta_2)$. There are $2$ questions about this I would like to know the answer to and how you would find it.

1. At what point $(x,y)$ does this loop have the intersection shown in the image, and
2. What is the area of the loop within this equation? I understand the equation is $1/2$ of the integral of $r^2 d\theta$ from $\theta_1$ to $\theta_2$, I just don't know how the get the $\theta_1$ and $\theta_2$ for this equation.

Answer 1

Hint

The point of intersection represents two points with the same absolute value of $r$, but with different values of $\theta$ seperated by an integer multiple of $\pi$. Since we are restricted to $[0,2\pi]$, this would be $\pi$

$$r_1 = \ln\theta_1$$

$$r_2 = \ln\theta_2 = \ln(\theta_1 + \pi)$$

Since log is an increasing function, we have $r_2 = -r_1, r_1 < 0$

$$\ln(\theta_1) + \ln(\theta_1 + \pi) = 0$$

$$\implies \ln(\theta_1(\theta_1 + \pi)) = 0$$

$$\implies \theta_1 = \frac{1}{\theta_1 + \pi}$$