Question

which of the following shows the graph of $y = 2\ln x$?

Explanation:

Step1: Analyze the function \( y = 2\ln x \)

The domain of \( y=\ln x \) is \( x>0 \), so \( y = 2\ln x \) is also defined for \( x>0 \). At \( x = 1 \), \( y=2\ln(1)=0 \) (since \( \ln(1) = 0 \)). The function \( y = \ln x \) passes through \( (1,0) \) and is increasing, and \( y = 2\ln x \) is a vertical stretch of \( y=\ln x \), still increasing, defined for \( x>0 \), passing through \( (1,0) \).

Step2: Analyze the graphs

The first graph (top) has a part where \( x<0 \), but \( y = 2\ln x \) is not defined for \( x\leq0 \), so it's incorrect. Wait, maybe the second graph (bottom) is the correct one? Wait, let's re - check. Wait, the function \( y = 2\ln x \): when \( x = 1 \), \( y = 0 \); as \( x
ightarrow0^{+} \), \( y
ightarrow-\infty \); as \( x
ightarrow+\infty \), \( y
ightarrow+\infty \), and it's increasing. The first graph (top) has a point at \( x = 0 \) and \( x<0 \), which is invalid. The second graph (bottom) is defined for \( x>0 \), starts from below (as \( x
ightarrow0^{+} \), \( y
ightarrow-\infty \)) and increases, passing through \( x = 1 \) (where \( y = 0 \))? Wait, maybe the second graph (the lower one) is the correct one. Wait, the first graph (top) has a curve that goes through \( x = 0 \), but \( \ln x \) is not defined at \( x = 0 \). So the correct graph should be the one where the curve is only for \( x>0 \), starts from the bottom (as \( x
ightarrow0^{+} \), \( y
ightarrow-\infty \)) and increases, passing through \( (1,0) \). So the second graph (the lower one) is the correct one. But since the user's image shows two graphs, the correct one is the lower graph (the one with \( y \) - axis and \( x>0 \) curve). Wait, maybe the first graph in the image (the top one) has an error in the \( x<0 \) part, so the correct graph is the second one (the lower one).

Answer:

The lower graph (the one below the first graph, the second graph in the image)