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**"3. Write the equation of a tangent line to the function evaluated at the given point. f(x) = 2 ln(x) at x = 3. What is the equation of the tangent line in y = mx + b form?"**

### Solution:

To find the equation of the tangent line, we need to follow these steps:

1. **Find the derivative of the function** \( f(x) = 2 \ln(x) \) to determine the slope of the tangent line.
   
   The derivative \( f'(x) \) represents the slope of the tangent line at any point \( x \).
   
   \[
   f'(x) = \frac{d}{dx} [2 \ln(x)] = \frac{2}{x}
   \]

2. **Evaluate the derivative at \( x = 3 \)** to find the slope of the tangent line at that specific point.

   \[
   f'(3) = \frac{2}{3}
   \]

   So, the slope \( m \) of the tangent line at \( x = 3 \) is \( \frac{2}{3} \).

3. **Find the function value at \( x = 3 \)** to determine the point on the curve where the tangent line touches.

   \[
   f(3) = 2 \ln(3)
   \]

   Therefore, the point is \( (3, 2\ln(3)) \).

4. **Use the point-slope form of a line** to find the equation of the tangent line. The point-slope form is given by:

   \[
   y - y_1 = m(x - x_1)
   \]

   Substituting \( m = \frac{2}{3} \), \( x_1 = 3 \), and \( y_1 = 2 \ln(3) \):

   \[
   y - 2 \ln(3) = \frac{2}{3}(x - 3)
   \]

5. **Solve for \( y \) to get the equation in the form \( y = mx + b \)**:

   \[
   y = \frac{2}{3}x + 2 \ln(3) - 2
   \]

Thus, the equation of the tangent line in \( y = mx + b \) form is:

\[
y = \frac{2}{3}x + 2 \ln(3) - 2
\]

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Do you want details or have any questions? 

Here are five related questions to consider:
1. How do you find the derivative of more complex logarithmic functions?
2. What is the significance of the point-slope form of a line?
3. How would the tangent line change if the point were different, say at \( x = 1 \)?
4. What are some real-world applications of finding the tangent line to a curve?
5. Can you explain why the tangent line is an approximation of the function near the point of tangency?

**Tip:** Understanding how to find the derivative is key to solving problems involving the slope of a tangent line. Practice differentiating various types of functions to improve your skills.

Write the equation of a tangent line to the function evaluated at the given point. f(x) = 2 ln(x) at x = 3. What is the equation of the tangent line in y = mx + b form?

Learn how to find the equation of a tangent line to the function f(x) = 2 ln(x) at x = 3. This solution includes differentiation techniques and the use of the point-slope form. Suitable for students in Grades 11-12.

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