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To find the equation of the tangent line to the graph of \( y = 4 \ln(x) \) at \( x = 4 \), we can follow these steps:

### Step 1: Find the derivative of the function.
The function is:
\[
y = 4 \ln(x)
\]
The derivative, \( y' \), gives the slope of the tangent line at any point \( x \). Using the derivative of the natural logarithm function, \( \frac{d}{dx}[\ln(x)] = \frac{1}{x} \), we get:
\[
y' = 4 \cdot \frac{1}{x} = \frac{4}{x}
\]

### Step 2: Evaluate the derivative at \( x = 4 \).
This gives the slope of the tangent line at the point where \( x = 4 \):
\[
y'(4) = \frac{4}{4} = 1
\]
So, the slope of the tangent line is \( m = 1 \).

### Step 3: Find the point on the curve at \( x = 4 \).
Substitute \( x = 4 \) into the original equation to find the corresponding \( y \)-value:
\[
y(4) = 4 \ln(4)
\]
Using the approximation \( \ln(4) \approx 1.386 \), we get:
\[
y(4) = 4 \times 1.386 = 5.544
\]
So, the point of tangency is \( (4, 5.544) \).

### Step 4: Use the point-slope form of the equation of a line.
The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Substituting the slope \( m = 1 \) and the point \( (4, 5.544) \):
\[
y - 5.544 = 1(x - 4)
\]
Simplifying:
\[
y = x - 4 + 5.544
\]
\[
y = x + 1.544
\]

### Final Answer:
The equation of the tangent line is:
\[
y = x + 1.544
\]

Would you like further details or have any questions about this solution? 

Here are some related questions to consider:
1. How would the tangent line change if the function were \( y = 4 \ln(2x) \)?
2. Can you find the second derivative of the function \( y = 4 \ln(x) \)?
3. What is the geometric meaning of the slope of the tangent line?
4. How does the natural logarithmic function behave for values of \( x \) close to 0?
5. How would the equation change if we considered a different point, say \( x = 5 \)?

**Tip:** When working with logarithmic functions, always ensure the argument inside the logarithm is positive to avoid undefined values.

Find the equation of the line tangent to the graph of y = 4 ln(x) at x = 4.

Learn how to find the equation of the tangent line to the function y = 4 ln(x) at x = 4. This calculus problem involves finding the derivative of logarithmic functions, evaluating the slope, and using the point-slope formula to find the tangent line. Suitable for students in Grades 11-12.

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