find the eqution of the line tangent to the graph of y=4lnx at x=3

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.

Find the equation of the line tangent to the graph of y=4lnx at x=3.

Learn how to find the equation of the tangent line to the curve y = 4ln(x) at the point x = 3. This solution explains the differentiation of logarithmic functions and uses the point-slope form to write the equation of the tangent line. Suitable for high school calculus or college students.

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