If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?
1. **Identify the expressions for \( x \) and \( y \):**
   \[
   x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \cdots \cos 89^\circ
   \]
   \[
   y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \cdots \cos 86^\circ
   \]

2. **Express \( y \) in terms of sine using the identity \(\cos \theta = \sin (90^\circ - \theta)\):**
   \[
   y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \cdots \cos 86^\circ = \sin 88^\circ \sin 84^\circ \sin 80^\circ \cdots \sin 4^\circ
   \]

3. **Use the sine double angle formula \(\sin 2\theta = 2 \sin \theta \cos \theta\) in reverse:**
   \[
   y = (2 \sin 44^\circ \cos 44^\circ)(2 \sin 42^\circ \cos 42^\circ) \cdots (2 \sin 2^\circ \cos 2^\circ)
   \]
   \[
   y = 2^{22} \sin 44^\circ \sin 42^\circ \cdots \sin 2^\circ \cos 44^\circ \cos 42^\circ \cdots \cos 2^\circ
   \]

4. **Combine the sine terms:**
   \[
   y = 2^{22} \sin 2^\circ \sin 4^\circ \cdots \sin 88^\circ
   \]

5. **Apply the double angle formula again:**
   \[
   y = 2^{22} \cdot 2^{22} \sin 1^\circ \sin 2^\circ \cdots \sin 44^\circ \sin 46^\circ \sin 47^\circ \cdots \sin 89^\circ
   \]
   \[
   y = 2^{44} \sin 1^\circ \sin 2^\circ \cdots \sin 44^\circ \sin 46^\circ \sin 47^\circ \cdots \sin 89^\circ
   \]

6. **Use \(\sin \theta = \cos (90^\circ - \theta)\) again:**
   \[
   y = \frac{2^{44} \cos 1^\circ \cos 2^\circ \cdots \cos 89^\circ}{\sin 45^\circ}
   \]
   \[
   \sin 45^\circ = \frac{1}{\sqrt{2}}
   \]
   \[
   y = 2^{44} \cdot \sqrt{2} \cdot \cos 1^\circ \cos 2^\circ \cdots \cos 89^\circ
   \]

7. **Divide \( y \) by \( x \):**
   \[
   \frac{y}{x} = \frac{2^{44} \cdot \sqrt{2} \cdot \cos 1^\circ \cos 2^\circ \cdots \cos 89^\circ}{\cos 1^\circ \cos 2^\circ \cdots \cos 89^\circ} = 2^{44} \cdot \sqrt{2}
   \]
   \[
   \frac{y}{x} = 2^{44} \cdot 2^{1/2} = 2^{44.5}
   \]

8. **Calculate \(\frac{2}{7} \log_2 \frac{y}{x}\):**
   \[
   \frac{2}{7} \log_2 (2^{44.5}) = \frac{2}{7} \cdot 44.5 = \frac{89}{7} \approx 12.714
   \]

9. **Find the nearest integer:**
   \[
   \text{The nearest integer to } 12.714 \text{ is } 13.
   \]

The final answer is \(\boxed{13}\)