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The given expression involves a limit:

\[
\lim_{x \to 0} \frac{\ln(\cos(5x))}{\ln(\cos(4x))}
\]

To evaluate this limit, we can use series expansion or L'Hopital's Rule, as the direct substitution of \(x = 0\) results in an indeterminate form \(\frac{0}{0}\).

### Steps for solving:

1. **Apply Taylor series expansion for small \(x\):**

   For small values of \(x\), the Taylor expansion for \(\cos(x)\) around \(x = 0\) is:
   \[
   \cos(x) \approx 1 - \frac{x^2}{2}
   \]

   Applying this to \(\cos(5x)\) and \(\cos(4x)\), we get:
   \[
   \cos(5x) \approx 1 - \frac{(5x)^2}{2} = 1 - \frac{25x^2}{2}
   \]
   \[
   \cos(4x) \approx 1 - \frac{(4x)^2}{2} = 1 - \frac{16x^2}{2}
   \]

2. **Logarithmic approximation:**

   Using the approximation \(\ln(1 - u) \approx -u\) for small \(u\), we can approximate the logarithms:
   \[
   \ln(\cos(5x)) \approx -\frac{25x^2}{2}, \quad \ln(\cos(4x)) \approx -\frac{16x^2}{2}
   \]

3. **Simplifying the ratio:**

   Now, substitute these approximations into the original limit expression:
   \[
   \lim_{x \to 0} \frac{\ln(\cos(5x))}{\ln(\cos(4x))} \approx \lim_{x \to 0} \frac{-\frac{25x^2}{2}}{-\frac{16x^2}{2}} = \frac{25}{16}
   \]

### Conclusion:
The value of the limit is:
\[
\boxed{\frac{25}{16}}
\]

Let me know if you would like further details or have any questions!

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Here are five related questions to explore:
1. How do Taylor series expansions work for trigonometric functions?
2. Can we use L'Hopital's rule in this case, and how would it apply?
3. How does the approximation \(\ln(1 - u) \approx -u\) work for small values of \(u\)?
4. What are other common limits involving trigonometric functions?
5. Can you extend this problem to higher powers of \(x\) in the cosine expansions?

**Tip**: Taylor expansions are a powerful tool for approximating functions near specific points, particularly when dealing with indeterminate forms in limits.

Evaluate the limit: lim (x -> 0) [ln(cos(5x)) / ln(cos(4x))]

Learn how to evaluate the limit of ln(cos(5x)) / ln(cos(4x)) as x approaches 0 using Taylor series expansion and logarithmic approximations. Understand the application of L'Hopital's Rule and series expansions in solving indeterminate forms. Suitable for students in Grades 11-12.

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