To find the value of \( x \) where the two functions intersect, we set the equations equal to each other:

\[
1.0545^x = \frac{1.057^x}{1280}
\]

This can be rewritten as:

\[
1280 \times 1.0545^x = 1.057^x
\]

To solve for \( x \), we take the logarithm of both sides:

\[
\log(1280) + x \log(1.0545) = x \log(1.057)
\]

Now, isolate \( x \):

\[
\log(1280) = x \log(1.057) - x \log(1.0545)
\]

\[
\log(1280) = x \left( \log(1.057) - \log(1.0545) \right)
\]

\[
x = \frac{\log(1280)}{\log(1.057) - \log(1.0545)}
\]

I'll calculate the exact value of \( x \).The value of \( x \) where the two functions intersect is approximately \( 3021.39 \).

Would you like to explore anything further regarding this result?

### Relative Questions:
1. How does the intersection point change if the base values (1.0545 or 1.057) are altered?
2. What does the intersection point signify in the context of these two functions?
3. Can you find the value of \( y \) at the intersection point?
4. How would the intersection change if the multiplier (1280) was different?
5. What does the solution imply about the behavior of these functions as \( x \) increases?

**Tip:** When solving logarithmic equations, remember that the logarithm is an increasing function, which ensures that each base corresponds to a unique solution for \( x \).

Explore how to find the intersection point of exponential functions with a detailed step-by-step solution. This problem involves equations with exponential functions and logarithms. Suitable for advanced high school students.

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