We realize the following relationship exists between dependent and independent variables: y = a bx.

To be able to fit a linear regression model, we transform the function and get the following linear function y*= b0 + b1 x.

After determining what transformation was used, answer the following questions.                                                     What is the value of b1?

Group of answer choices

exp(b)

1/b

b

Ln(b)        

The given relationship is \( y = a \cdot b^x \), which is an exponential model. To fit this into a linear regression model, we apply a logarithmic transformation to both sides. Let's transform the equation step by step.

1. Start with the original equation:
   \[
   y = a \cdot b^x
   \]
   
2. Apply the natural logarithm (\( \ln \)) to both sides:
   \[
   \ln(y) = \ln(a \cdot b^x)
   \]

3. Using logarithmic properties, split the terms:
   \[
   \ln(y) = \ln(a) + \ln(b^x)
   \]

4. Further simplify using the property \( \ln(b^x) = x \cdot \ln(b) \):
   \[
   \ln(y) = \ln(a) + x \cdot \ln(b)
   \]

5. Now, let \( y^* = \ln(y) \) and \( b_0 = \ln(a) \), and rewrite the equation:
   \[
   y^* = b_0 + b_1 \cdot x
   \]

Here, \( b_1 \) is the slope of the linearized model, and we see from the transformed equation that \( b_1 = \ln(b) \).

Thus, the correct answer is:

**\( \ln(b) \)**

### Do you want more details or have any other questions?

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Here are five related questions:

1. What is the role of \( b_0 \) in this transformed linear model?
2. How can we determine \( a \) after performing the logarithmic transformation?
3. What does the slope \( b_1 = \ln(b) \) represent in the context of exponential growth?
4. How would the transformation change if the base of the exponent were different from \( e \)?
5. How does this transformation affect the interpretation of residuals in linear regression?

**Tip:** Always remember that when fitting models, transformations like logarithms help linearize non-linear relationships, making it easier to apply linear regression techniques.

We realize the following relationship exists between dependent and independent variables: y = a bx. To be able to fit a linear regression model, we transform the function and get the following linear function y* = b0 + b1 x. After determining what transformation was used, answer the following questions. What is the value of b1? Group of answer choices: exp(b), 1/b, b, Ln(b)

Learn how to transform an exponential function y = a * b^x into a linear regression model using logarithmic transformation. This detailed solution explains the transformation process and how to find the value of b1 as ln(b). Suitable for Grades 10-12.

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