Let's solve the problem step-by-step. Here's what we know:

1. Two Circles:

   • Circle with center $O$ and radius $r$.
   • Circle with center $P$ and radius $s$.

2. Intersection Points:

   • The two circles intersect at points $A$ and $B$.
   • The line segment $OP$ intersects the circle with center $O$ at point $C$.

3. Given Lengths:

   • $OP = 6$
   • $OC = 2$

4. Angle Information:

   • $\angle AOP = 60^\circ$.

5. Objective:

   • Find the value of $\log_e \left(\frac{r}{s}\right)$.

Solution

Step 1: Determine $r$ Using Triangle $\triangle OCP$

Since $OP = 6$ and $OC = 2$, we can use the relationship in triangle $OCP$.

From the law of cosines in $\triangle OCP$: $OP^2 = OC^2 + CP^2 - 2 \cdot OC \cdot CP \cdot \cos(60^\circ)$ Since $\cos(60^\circ) = \frac{1}{2}$, substitute values: $6^2 = 2^2 + CP^2 - 2 \cdot 2 \cdot CP \cdot \frac{1}{2}$ $36 = 4 + CP^2 - 2 \cdot CP$