We are given the following problem:

• A tangent line $\overline{AB}$ is drawn to the curve $y = \ln\left(\frac{x}{2}\right)$ at the point $C(1, y)$.
• The line intersects the x-axis at point $A$ and the y-axis at point $B$.
• We are tasked with finding the length of the tangent line $\overline{AB}$.

Steps to solve:

1. Find the derivative of the function $y = \ln\left(\frac{x}{2}\right)$. $\frac{dy}{dx} = \frac{1}{x}$ This gives us the slope of the tangent line at any point on the curve.

2. Find the slope of the tangent line at point $C(1, y)$: At $C(1, y)$, the slope of the tangent line is: $\frac{dy}{dx}\Big|_{x=1} = \frac{1}{1} = 1$ So, the slope of the tangent line at point $C(1, 0)$ is 1.

3. Equation of the tangent line: The equation of the tangent line in point-slope form is: $y - y_1 = m(x - x_1)$ Substituting $m = 1$ and $(x_1, y_1) = (1, \ln\left(\frac{1}{2}\right)) = (1, -\ln 2)$: $y - (-\ln 2) = 1(x - 1)$ $y + \ln 2 = x - 1$ $y = x - 1 - \ln 2$ This is the equation of the tangent line.

4. Find the x-intercept (point A): To find the x-intercept, set $y = 0$ and solve for $x$: $0 = x - 1 - \ln 2$ $x = 1 + \ln 2$ So, point $A$ is $(1 + \ln 2, 0)$.

5. Find the y-intercept (point B): To find the y-intercept, set $x = 0$ and solve for $y$: $y = 0 - 1 - \ln 2$ $y = -1 - \ln 2$ So, point $B$ is $(0, -1 - \ln 2)$.

6. Calculate the length of $\overline{AB}$: The length of the line segment $\overline{AB}$ can be calculated using the distance formula: $\text{Length of } \overline{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting the coordinates of points $A(1 + \ln 2, 0)$ and $B(0, -1 - \ln 2)$: $\text{Length of } \overline{AB} = \sqrt{(0 - (1 + \ln 2))^2 + ((-1 - \ln 2) - 0)^2}$ $= \sqrt{(1 + \ln 2)^2 + (-1 - \ln 2)^2}$ $= \sqrt{(1 + \ln 2)^2 + (1 + \ln 2)^2}$ $= \sqrt{2(1 + \ln 2)^2}$ $= \sqrt{2}(1 + \ln 2)$ Hence, the length of $\overline{AB}$ is: $\boxed{\sqrt{2}(1 + \ln 2) \, \text{units}}$

Would you like further clarification on any step? Here are five related questions:

1. How would the equation of the tangent line change if the point of tangency were different?
2. Can we find the length of $\overline{AB}$ using a different method?
3. How do we interpret the natural logarithmic function geometrically on this curve?
4. What happens to the slope of the tangent line as $x \to \infty$?
5. What role does the point $C(1, y)$ play in determining the tangent's properties?

Tip: Always differentiate the function first when dealing with tangent lines to a curve!