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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:
\[
6^2 = 2^2 + CP^2 - 2 \cdot 2 \cdot CP \cdot \frac{1}{2}
\]
\[
36 = 4 + CP^2 - 2 \cdot CP
\]


A circle with center O and radius r intersects another circle with center P and radius s at two points A and B. The line segment OP intersects the circle with center O at point C. If the length of OP is 6, the length of OC is 2, and the angle ∠AOP is 60°, find the value of log_e (r/s).

Solve a geometry problem involving two intersecting circles with given lengths and angles. Use the law of cosines to find log_e(r/s). Suitable for advanced high school students, covering trigonometry and logarithmic concepts.

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