The curve with equation y = ln x is transformed by a stretch parallel to the x-axis with scale factor 2. Find the equation of the transformed curve.

Steps to Solve

1. Understand the Stretch Transformation A stretch parallel to the x-axis with a scale factor of 2 means that every x-coordinate of the curve will be multiplied by 2.

2. Apply the Transformation to the Original Equation The original equation is given by $y = \ln x$.

To apply the stretch, we replace $x$ with $\frac{x}{2}$ in the equation. This gives us the new equation: $$ y = \ln\left(\frac{x}{2}\right) $$

3. Simplify the Transformed Equation Using logarithmic properties, we can simplify the transformed equation: $$ y = \ln\left(\frac{x}{2}\right) = \ln x - \ln 2 $$

Since we need to express the transformed equation in the form provided in the options, we can express it as: $$ y = \ln x - \ln 2 $$

4. Identify the Correct Answer from the Given Options Given the options to choose from, we find the one that matches the transformed equation.

The correct option from the list is: $$ y = \ln\left(\frac{x}{2}\right) $$