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The image contains two problems related to curve fitting and determining equations from data points. Here's a breakdown of the questions:

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### Problem 1:
You are asked to **find a hyperbola equation \( x y = a \)** that best fits the given data points:

| \( x \)   | 0.5  | 1.0  | 2.0  | 3.0  | 5.0  |
|----------|-------|------|------|------|------|
| \( y \)  | 62   | 28   | 17   | 9.0  | 5.0  |

The hyperbola equation is given as \( xy = a \), where \( a \) is a constant. We need to find the value of \( a \) that satisfies this equation for the given \( x \) and \( y \) values.

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### Problem 2:
You are asked to find the value of \( a \) for the logarithmic equation **\( y = a \ln(x) \)** that best fits the following data points:

| \( x \)   | 0.3  | 0.9  | 1.7  | 2.5  | 4.0  | 10.0  |
|----------|-------|------|------|------|------|-------|
| \( y \)  | -7.54 | -0.57 | 3.7  | 4.61 | 8.27 | 12.2  |

You are to determine the value of \( a \) such that the equation \( y = a \ln(x) \) fits the given points as closely as possible.

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I'll now calculate the solutions for both problems. Let's begin with Problem 1.### Problem 1 Solution:
The hyperbola equation is \( xy = a \), and after calculating for the given points:

- For \( x = 0.5, y = 62 \), \( a = 31 \)
- For \( x = 1.0, y = 28 \), \( a = 28 \)
- For \( x = 2.0, y = 17 \), \( a = 34 \)
- For \( x = 3.0, y = 9.0 \), \( a = 27 \)
- For \( x = 5.0, y = 5.0 \), \( a = 25 \)

The average value of \( a \), which best fits the data, is **29.0**. Thus, the hyperbola that best fits the data is approximately:

\[
xy = 29
\]

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Next, I'll solve Problem 2 by finding the best value of \( a \) for the logarithmic model \( y = a \ln(x) \).### Problem 2 Solution:
For the logarithmic equation \( y = a \ln(x) \), the best fit for the data gives:

\[
a \approx 5.60
\]

Thus, the equation that best fits the data is:

\[
y = 5.60 \ln(x)
\]

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Would you like further details or explanations on the steps? Here are some related questions for practice:

1. How would you solve for \( x \) given the hyperbola equation \( xy = 29 \) and a specific value of \( y \)?
2. What method can be used to approximate \( a \) in nonlinear equations like \( y = a \ln(x) \)?
3. How can logarithmic regression be applied to different types of data sets?
4. What are some real-world applications of the equation \( y = a \ln(x) \)?
5. How does the curve fitting process change if the data contains outliers?

**Tip:** When fitting curves to data, always check the residuals (the difference between observed and fitted values) to evaluate the goodness of the fit.

1. Buatlah hiperbola persamaan x * y = a yang terbaik untuk himpunan nilai berikut: 
| x: 0.5, 1.0, 2.0, 3.0, 5.0 |
| y: 62, 28, 17, 9.0, 5.0 |

2. Nilai-nilai x dan y yang dihasilkan oleh persamaan y = a * ln(x) berikut: 
| x: 0.3, 0.9, 1.7, 2.5, 4.0, 10.0 |
| y: -7.54, -0.57, 3.7, 4.61, 8.27, 12.2 |
Tentukan nilai a agar dapat menghasilkan kurva terbaik.

Solve the curve fitting problems using hyperbola and logarithmic regression. Find the best fit for data points using equations xy = a and y = a ln(x). Includes step-by-step solutions.

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