Questions: Find the equation of the line tangent to the graph of f(x) = (ln x)^2 at x = 8.

Solution Steps

To find the equation of the tangent line to the graph of \( f(x) = (\ln x)^2 \) at \( x = 8 \), we need to follow these steps:

1. Compute the derivative \( f'(x) \) to find the slope of the tangent line at \( x = 8 \).
2. Evaluate \( f(8) \) to find the y-coordinate of the point of tangency.
3. Use the point-slope form of the equation of a line to write the equation of the tangent line.

To find the slope of the tangent line at \( x = 8 \), we first compute the derivative of the function \( f(x) = (\ln x)^2 \). The derivative is given by: \[ f'(x) = \frac{2 \ln x}{x} \]

Next, we evaluate the derivative at \( x = 8 \): \[ f'(8) = \frac{2 \ln(8)}{8} \approx 0.5199 \] We also evaluate the function at \( x = 8 \): \[ f(8) = (\ln(8))^2 \approx 4.3241 \]

Using the point-slope form of the equation of a line, we can express the tangent line as: \[ y - f(8) = f'(8)(x - 8) \] Rearranging this to slope-intercept form \( y = mx + b \), we find the y-intercept \( b \): \[ b = f(8) - f'(8) \cdot 8 \approx 0.1652 \]

Final Answer