A circle with center 
O
O and radius 
r
r intersects another circle with center 
P
P and radius 
s
s at two points 
A
A and 
B
B. The line segment 
O
P
OP intersects the circle with center 
O
O at point 
C
C. If the length of 
O
P
OP is 
6
6, the length of 
O
C
OC is 
2
2, and the angle 
∠
A
O
P
∠AOP is 
6
0
∘
60 
∘
 , find the value of 
log
⁡
e
(
r
s
)
log 
e
​
 ( 
s
r
​
 ).

To solve this problem, we can analyze the given geometry and use trigonometric relationships.

### Step-by-Step Solution

1. **Given Information**:
   - \( OP = 6 \)
   - \( OC = 2 \)
   - \( \angle AOP = 60^\circ \)

2. **Relation Between \( OC \) and \( OP \)**:
   Since \( OC = 2 \), \( C \) is a point on the circle with center \( O \) and radius \( r \). Therefore, \( OC = r = 2 \).

3. **Using the Cosine Rule in Triangle \( OCP \)**:
   In \( \triangle OCP \), we know:
   - \( OP = 6 \)
   - \( OC = r = 2 \)
   - \( \angle OCP = 60^\circ \)

   To find \( s = CP \), we use the Cosine Rule:
   \[
   CP^2 = OP^2 + OC^2 - 2 \cdot OP \cdot OC \cdot \cos(60^\circ)
   \]
   Substituting the values:
   \[
   s^2 = 6^2 + 2^2 - 2 \cdot 6 \cdot 2 \cdot \cos(60^\circ)
   \]
   \[
   = 36 + 4 - 2 \cdot 6 \cdot 2 \cdot \frac{1}{2}
   \]
   \[
   = 40 - 12 = 28
   \]
   \[
   s = \sqrt{28} = 2\sqrt{7}
   \]

4. **Calculate \( \ln\left(\frac{s}{r}\right) \)**:
   We need to find \( \ln\left(\frac{s}{r}\right) \).
   
   Since \( r = 2 \) and \( s = 2\sqrt{7} \), we have:
   \[
   \frac{s}{r} = \frac{2\sqrt{7}}{2} = \sqrt{7}
   \]
   Thus,
   \[
   \ln\left(\frac{s}{r}\right) = \ln\left(\sqrt{7}\right)
   \]
   Using the property \( \ln(\sqrt{a}) = \frac{1}{2} \ln(a) \), we get:
   \[
   \ln\left(\sqrt{7}\right) = \frac{1}{2} \ln(7)
   \]

### Final Answer:
\[
\ln\left(\frac{s}{r}\right) = \frac{1}{2} \ln(7)
\]

Would you like further clarification on any of the steps?

---

### Follow-up Questions:
1. How does the cosine rule apply in non-right triangles?
2. What other trigonometric methods can help solve geometric problems involving circles?
3. Can the solution be extended if the angle \( \angle AOP \) were different?
4. What if the radius \( s \) were given? How would that simplify the problem?
5. How is logarithmic simplification applied in different mathematical contexts?

**Tip:** Always check if given angles allow using symmetry or special triangle properties to simplify complex problems.

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).

Explore how to find the value of log_e(r/s) for two intersecting circles with centers O and P, given OP = 6, OC = 2, and angle ∠AOP = 60°. This solution uses trigonometric and logarithmic concepts, ideal for students in Grades 10-12.

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