Three logarithmic functions are graphed on a coordinate grid.Three logarithmic functions are graphed on a coordinate grid. All are increasing and pass through the point left-parenthesis 1 comma 0 right-parenthesis. Between x equals 0 and x equals 1, the curve labeled c is on top, the curve labeled b is in the middle and the curve labeled a is on the bottom. For all x-values greater than 1, the curve labeled a is on top, the curve labeled b is in the middle, and the curve labeled c is on the bottom.

Question

Question
Use the graph to answer the question.
Match the three graphs to the functions they represent: f left parenthesis x right parenthesis equals log subscript 7 baseline x, g left parenthesis x right parenthesis equals log subscript 3 baseline x, h left parenthesis x right parenthesis equals log subscript 2 baseline x.
(1 point)
Responses

a. f left parenthesis x right parenthesis equals log subscript 7 baseline x b. g left parenthesis x right parenthesis equals log subscript 3 baseline x c. fhleft parenthesis x right parenthesis equals log subscript 2 baseline x
Image with alt text: a. f left parenthesis x right parenthesis equals log subscript 7 baseline x b. g left parenthesis x right parenthesis equals log subscript 3 baseline x c. fhleft parenthesis x right parenthesis equals log subscript 2 baseline x

a. h left parenthesis x right parenthesis equals log subscript 2 baseline x b. g left parenthesis x right parenthesis equals log subscript 3 baseline x c. f left parenthesis x right parenthesis equals log subscript 7 baseline x
Image with alt text: a. h left parenthesis x right parenthesis equals log subscript 2 baseline x b. g left parenthesis x right parenthesis equals log subscript 3 baseline x c. f left parenthesis x right parenthesis equals log subscript 7 baseline x

a. g left parenthesis x right parenthesis equals log subscript 3 baseline x b. f left parenthesis x right parenthesis equals log subscript 7 baseline x c. h left parenthesis x right parenthesis equals log subscript 2 baseline x
Image with alt text: a. g left parenthesis x right parenthesis equals log subscript 3 baseline x b. f left parenthesis x right parenthesis equals log subscript 7 baseline x c. h left parenthesis x right parenthesis equals log subscript 2 baseline x

a. f left parenthesis x right parenthesis equals log subscript 7 baseline x b. h left parenthesis x right parenthesis equals log subscript 2 baseline x c. g left parenthesis x right parenthesis equals log subscript 3 baseline x
Image with alt text: a. f left parenthesis x right parenthesis equals log subscript 7 baseline x b. h left parenthesis x right parenthesis equals log subscript 2 baseline x c. g left parenthesis x right parenthesis equals log subscript 3 baseline x
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Answer 1

To match the logarithmic functions to the curves based on the given characteristics:

1. Behavior between \( x = 0 \) and \( x = 1 \):

   • Curve \( c \) is on top, followed by curve \( b \), and curve \( a \) is at the bottom.
   • Since logarithmic functions increase less steeply for larger bases, the highest logarithmic function (most increasing) between \( 0 \) and \( 1 \) will correspond to the smallest base, which is \( \log_2(x) \). Therefore, curve \( c \) corresponds to \( h(x) = \log_2(x) \).

2. Behavior for \( x > 1 \):

   • Here, curve \( a \) is on top, followed by curve \( b \), and curve \( c \) is at the bottom.
   • As \( x \) increases, logarithmic functions with larger bases grow faster. Thus, the curve that is at the top in this region will correspond to the largest base, which is \( \log_7(x) \). Consequently, curve \( a \) matches with \( f(x) = \log_7(x) \).
   • This leaves the middle curve, \( b \), which corresponds to \( g(x) = \log_3(x) \).

Given this reasoning, we can summarize the matches as follows:

• Curve \( a \): \( f(x) = \log_7(x) \)
• Curve \( b \): \( g(x) = \log_3(x) \)
• Curve \( c \): \( h(x) = \log_2(x) \)

Thus, the correct response is:

a. \( f(x) = \log_7(x) \), b. \( g(x) = \log_3(x) \), c. \( h(x) = \log_2(x) \).