Find a logarithmic equation that relates y and x. (Round any numeric values to three decimal places.)
ln(y) = 
  
x	1	2	3	4	5	6
y	1.500	1.306	1.204	1.137	1.087	1.048


To find a logarithmic equation that relates \( y \) and \( x \), we will start by examining the relationship between \( \ln(y) \) and \( x \). The general form of a logarithmic equation we are looking for is:

\[
\ln(y) = ax + b
\]

where \( a \) and \( b \) are constants to be determined through linear regression, using the given data points for \( x \) and \( \ln(y) \).

### Step 1: Calculate \( \ln(y) \) values
Using the \( y \) values provided, we calculate \( \ln(y) \) for each:

1. \( y = 1.500 \Rightarrow \ln(1.500) \approx 0.405 \)
2. \( y = 1.306 \Rightarrow \ln(1.306) \approx 0.267 \)
3. \( y = 1.204 \Rightarrow \ln(1.204) \approx 0.186 \)
4. \( y = 1.137 \Rightarrow \ln(1.137) \approx 0.129 \)
5. \( y = 1.087 \Rightarrow \ln(1.087) \approx 0.083 \)
6. \( y = 1.048 \Rightarrow \ln(1.048) \approx 0.047 \)

So, we now have the following dataTo find a logarithmic equation that relates \( y \) and \( x \), we will start by examining the relationship between \( \ln(y) \) and \( x \). The general form of a logarithmic equation we are looking for is:

\[
\ln(y) = ax + b
\]

where \( a \) and \( b \) are constants to be determined through linear regression, using the given data points for \( x \) and \( \ln(y) \).

### Step 1: Calculate \( \ln(y) \) values
Using the \( y \) values provided, we calculate \( \ln(y) \) for each:

1. \( y = 1.500 \Rightarrow \ln(1.500) \approx 0.405 \)
2. \( y = 1.306 \Rightarrow \ln(1.306) \approx 0.267 \)
3. \( y = 1.204 \Rightarrow \ln(1.204) \approx 0.186 \)
4. \( y = 1.137 \Rightarrow \ln(1.137) \approx 0.129 \)
5. \( y = 1.087 \Rightarrow \ln(1.087) \approx 0.083 \)
6. \( y = 1.048 \Rightarrow \ln(1.048) \approx 0.047 \)

So, we now have the following 

| \( x \) | \( \ln(y) \) |
| ------- | ------------ |
| 1       | 0.405       |
| 2       | 0.267       |
| 3       | 0.186       |
| 4       | 0.129       |
| 5       | 0.083       |
| 6       | 0.047       |

### Step 2: Perform Linear Regression on \( x \) and \( \ln(y) \)
Using these pairs, we can calculate the best-fit line \( \ln(y) = ax + b \) by finding the values of \( a \) and \( b \) through regression.

Let's calculate these values:

- **Slope \( a \):** approximately \(-0.074\)
- **Intercept \( b \):** approximately \(0.463\)

### Step 3: Write the Equation
So, the logarithmic equation that relates \( y \) and \( x \) is:

\[
\ln(y) = -0.074x + 0.463
\]

### Step 4: Convert to Exponential Form (Optional)
If you'd like to express \( y \) in terms of \( x \), you can rewrite it as:

\[
y = e^{(-0.074x + 0.463)}
\]

Or, simplified:

\[
y \approx 1.589 \cdot e^{-0.074x}
\]

Would you like more details on any step? Here are some related questions:

1. How is linear regression applied to find the best-fit line?
2. What are the interpretations of the slope and intercept in this context?
3. How can we verify the accuracy of the equation using the given data points?
4. How would the result change if we rounded to fewer decimal places?
5. Can this method be extended to other types of functions?

**Tip:** When dealing with logarithmic or exponential relationships, transforming data (e.g., taking the natural logarithm) often simplifies the process of finding linear relationships.

Find a logarithmic equation that relates y and x. (Round any numeric values to three decimal places.)
ln(y) = 

x 1 2 3 4 5 6
y 1.500 1.306 1.204 1.137 1.087 1.048

Learn how to find a logarithmic equation relating y and x using linear regression. The problem involves calculating the natural logarithm of y and deriving the best-fit line, yielding the equation ln(y) = -0.074x + 0.463, which simplifies to y ≈ 1.589 * e^(-0.074x). Suitable for students in Grades 10-12.

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