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It seems like you’ve uploaded a math problem involving logarithmic regression. Based on the image, here’s a breakdown of the task:

### Part (a): Finding the Logarithmic Regression Model
The equation for logarithmic regression is:
\[
y = a + b \ln(x)
\]
From the solution shown:
- \( a = 1.195691839 \)
- \( b = 2.968695493 \)
Thus, the regression equation becomes:
\[
y = 1.195691839 + 2.968695493 \ln(x)
\]

### Part (b): Predicting \( y \) when \( x = 3 \)
To predict the value of \( y \), substitute \( x = 3 \) into the regression equation:
\[
y = 1.195691839 + 2.968695493 \ln(3)
\]
Using the natural logarithm of 3 (\(\ln(3) \approx 1.098612\)):
\[
y \approx 1.195691839 + 2.968695493 \cdot 1.098612
\]
\[
y \approx 4.4571371889234
\]
The prediction for \( y \) is approximately 4.4571.

Would you like further explanation on logarithmic regression or other related concepts?

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### Related Questions to Explore:
1. What are the applications of logarithmic regression in real-world scenarios?
2. How is the goodness-of-fit for a logarithmic regression model measured?
3. What would the graph of this logarithmic regression equation look like?
4. How do logarithmic and exponential regression models differ in application?
5. What happens to the regression equation if the data points are scaled differently?

**Tip:** Always check the residuals of a regression model to ensure it fits the data accurately.

The function f is increasing. The table below gives values for the function at selected values of x. A logarithmic regression y = a + b ln x is used to model the data. Find the logarithmic regression model and use it to predict f when x = 3.

Explore how to solve logarithmic regression equations like y = 1.1957 + 2.9687 ln(x). Learn how to predict the value of f when x = 3 using natural logarithms. Ideal for Grades 10-12.

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