Question

match the graphs that represent ( f(x) ) and ( g(x) ) to the table.

function	graph
( g(x) = log_2 x )

(images of four graphs are shown below the table, with x and y axes labeled.)

Explanation:

Step1: Analyze \( f(x) = 3\ln x \)

The function \( y = \ln x \) has a vertical asymptote at \( x = 0 \), passes through \( (1, 0) \), and is increasing (since the derivative \( \frac{1}{x}>0 \) for \( x>0 \)). The function \( f(x)=3\ln x \) is a vertical stretch of \( \ln x \) by a factor of 3. So it will grow faster than \( \ln x \) and pass through \( (1, 0) \), with a vertical asymptote at \( x = 0 \). Looking at the graphs, the fourth graph (rightmost) has a steeper increase, which matches the stretched logarithmic function.

Step2: Analyze \( g(x)=\log_{2}x \)

The function \( y = \log_{2}x \) has a vertical asymptote at \( x = 0 \), passes through \( (1, 0) \), and is increasing (derivative \( \frac{1}{x\ln 2}>0 \) for \( x>0 \)). The base - 2 logarithm grows slower than the natural logarithm (since \( \ln 2\approx0.693<1 \)). Among the remaining graphs, the second graph (from the left, after the first) has a more gradual increase compared to the fourth, and it passes through \( (1, 0) \) with a vertical asymptote at \( x = 0 \), which matches \( g(x)=\log_{2}x \). Wait, actually, let's re - check:

For \( f(x) = 3\ln x \): When \( x = e\), \( f(e)=3 \); when \( x = e^{2}\), \( f(e^{2}) = 6 \).

For \( g(x)=\log_{2}x \): When \( x = 2\), \( g(2)=1 \); when \( x = 4\), \( g(4)=2 \); when \( x = 8\), \( g(8)=3 \).

The first graph (leftmost) has a very slow increase, the second graph has a bit more, the third graph passes through (0,0) which is wrong (since \( \log_{2}0 \) is undefined), the fourth graph is steeper. Wait, maybe I made a mistake earlier. Let's list the key points:

• \( f(x)=3\ln x \): At \( x = e\approx2.718 \), \( f(x)=3\approx3 \); at \( x = 3 \), \( f(3)=3\ln 3\approx3\times1.0986 = 3.2958 \).

• \( g(x)=\log_{2}x \): At \( x = 2 \), \( g(x)=1 \); at \( x = 4 \), \( g(x)=2 \); at \( x = 8 \), \( g(x)=3 \).

The fourth graph (rightmost) has a steeper slope, so it's \( f(x)=3\ln x \). The second graph (from left, the one with a more gradual increase, passing through (1,0) and having a vertical asymptote at x = 0) is \( g(x)=\log_{2}x \). Wait, the third graph starts at (0,0) which is incorrect for logarithmic functions (since log functions are undefined at x = 0), so we can eliminate the third graph. The first graph is very flat, the second graph has a moderate increase, the fourth is steeper. So:

• \( f(x)=3\ln x \) corresponds to the Rightmost Graph (fourth graph).

• \( g(x)=\log_{2}x \) corresponds to the Second Graph (from the left).

Answer:

• For \( f(x) = 3\ln x \): The Rightmost Graph.
• For \( g(x)=\log_{2}x \): The Second Graph (from the left).