To find the equation of the line tangent to the graph of $y = 4 \ln(x)$ at $x = 3$, we follow these steps:

Step 1: Differentiate the function

The derivative of $y = 4 \ln(x)$ gives the slope of the tangent line at any point $x$. So, we calculate:

$\frac{dy}{dx} = \frac{d}{dx} [4 \ln(x)] = \frac{4}{x}$

Step 2: Evaluate the slope at $x = 3$

Now, substitute $x = 3$ into the derivative to find the slope of the tangent line at this point:

$\text{slope} = \frac{4}{3}$

Step 3: Find the point on the curve at $x = 3$

Substitute $x = 3$ into the original function $y = 4 \ln(x)$ to find the corresponding $y$-coordinate:

$y = 4 \ln(3)$

Using the approximation $\ln(3) \approx 1.0986$:

$y \approx 4 \times 1.0986 = 4.3944$

Thus, the point of tangency is $(3, 4.3944)$.

Step 4: Write the equation of the tangent line

Using the point-slope form of a line equation:

$y - y_1 = m(x - x_1)$

where $m = \frac{4}{3}$, $x_1 = 3$, and $y_1 = 4.3944$, we get:

$y - 4.3944 = \frac{4}{3}(x - 3)$

Simplifying the equation:

$y = \frac{4}{3}x - 4 + 4.3944$

$y = \frac{4}{3}x + 0.3944$

Thus, the equation of the tangent line is:

$y = \frac{4}{3}x + 0.3944$

Would you like more details or clarifications on any part of this process?

Related Questions:

1. How do you find the slope of a curve at any point using derivatives?
2. What is the general form of the equation of a tangent line?
3. How does the natural logarithmic function affect the shape of a curve?
4. Can the derivative of a logarithmic function be extended to include different bases?
5. What are other methods to approximate logarithmic values without a calculator?

Tip: Understanding derivatives is key to finding the slopes of tangent lines, which represent instantaneous rates of change.