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To solve this problem, we need to find the equation of the tangent line to the function \( f(x) = \ln(x^3) \) at \( x = 4 \).

### Steps:

1. **Rewrite the function**:
   \[
   f(x) = \ln(x^3) = 3 \ln(x)
   \]
   We applied the logarithmic identity \( \ln(a^b) = b \ln(a) \).

2. **Find the derivative \( f'(x) \)**:
   The derivative of \( f(x) = 3 \ln(x) \) with respect to \( x \) is:
   \[
   f'(x) = \frac{3}{x}
   \]
   This gives us the slope of the tangent line at any point \( x \).

3. **Evaluate \( f'(x) \) at \( x = 4 \)**:
   \[
   f'(4) = \frac{3}{4}
   \]
   So the slope of the tangent line at \( x = 4 \) is \( \frac{3}{4} \).

4. **Find \( f(4) \)**:
   Now, evaluate \( f(4) \):
   \[
   f(4) = 3 \ln(4)
   \]
   Using the approximation \( \ln(4) \approx 1.386 \), we get:
   \[
   f(4) = 3 \times 1.386 = 4.158
   \]

5. **Equation of the tangent line**:
   The equation of the tangent line is given by the point-slope form:
   \[
   y - f(4) = f'(4)(x - 4)
   \]
   Substituting the known values:
   \[
   y - 4.158 = \frac{3}{4}(x - 4)
   \]
   Simplifying:
   \[
   y = \frac{3}{4}(x - 4) + 4.158
   \]
   \[
   y = \frac{3}{4}x - 3 + 4.158
   \]
   \[
   y = \frac{3}{4}x + 1.158
   \]

Thus, the equation of the tangent line is:
\[
y = \frac{3}{4}x + 1.158
\]

Would you like further details or have any questions?

### Related Questions:
1. What are the general steps to find a tangent line to a function?
2. How does the logarithmic differentiation help in simplifying logarithmic functions?
3. Can you explain how point-slope form works for finding tangent lines?
4. What is the derivative of \( f(x) = \ln(x^3) \) without simplifying the logarithm?
5. How would the equation change if the function was \( f(x) = \ln(x^2) \)?

### Tip:
Always simplify logarithmic expressions using properties like \( \ln(a^b) = b \ln(a) \) before differentiating to make the process easier.

Find the equation of the tangent line to the function f(x) = ln(x^3) at x = 4.

Learn how to find the equation of the tangent line to the function f(x) = ln(x^3) at x = 4. This problem involves key calculus concepts, including logarithmic differentiation, point-slope form, and the logarithmic identity ln(a^b) = b ln(a). Suitable for students in Grades 10-12.

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