To find specific values of \(x\) and \(y\) for both experiments, we typically need additional information such as specific measurements or constraints. However, we can assume some values for \(x\) and calculate the corresponding \(y\) for both experiments.

Let's choose \(x = 10\) as a specific value and calculate \(y\) for both experiments.

### First Experiment
The relationship is given by:
\[ y = 10^4 x^2 \]

For \(x = 10\):
\[ y = 10^4 \cdot 10^2 \]
\[ y = 10^4 \cdot 10^2 = 10^6 \]

So, when \(x = 10\):
\[ y = 10^6 \]

### Second Experiment
The relationship is given by:
\[ y = 8x^{3 \log_{10} x - 1} \]

For \(x = 10\):
\[ n = 3 \log_{10} x - 1 = 3 \log_{10} 10 - 1 \]
\[ n = 3 \cdot 1 - 1 \]
\[ n = 3 - 1 = 2 \]

Thus, the equation becomes:
\[ y = 8 \cdot 10^2 \]
\[ y = 8 \cdot 100 = 800 \]

So, when \(x = 10\):
\[ y = 800 \]

Therefore, the specific values of \(x\) and \(y\) are:
- For the first experiment: \( (x, y) = (10, 10^6) \)
- For the second experiment: \( (x, y) = (10, 800) \)

### Questions for Further Exploration:

1. What happens to \(y\) if \(x = 20\) in both experiments?
2. How would the value of \(y\) change in the first experiment if \(k\) were \(10^5\)?
3. What is the relationship between \(x\) and \(y\) for a different value of \(n\) in the second experiment?
4. Can we derive a general formula for \(y\) in terms of \(x\) for different values of \(n\)?
5. How do these relationships change if \(x\) is less than 1?
6. What are the implications of the different exponents on the growth rate of \(y\) in each experiment?
7. How would the graph of \(y = 10^4 x^2\) compare to \(y = 8x^{3 \log_{10} x - 1}\)?
8. What are practical applications of these types of exponential relationships in real-world scenarios?

### Tip
When solving equations involving exponents and logarithms, verifying your calculations by substituting back into the original equation can help ensure accuracy.

Explore how to find specific values of x and y in mathematical experiments involving exponential relationships and logarithmic functions. Includes detailed calculations and explanations for two distinct experiments. Suitable for advanced high school students and undergraduates.

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