Question

use a graphing utility and the change-of-base property to graph the function
( y = log_7 x )
...
which graph is the correct graph of ( y = log_7 x )?
a
graph a

b
graph b

c
graph c

d
graph d

all graphs have the viewing window 0,10,1 by -3,3,1.

Explanation:

Step1: Recall log function properties

The function \( y = \log_{7}x \) is a logarithmic function with base \( 7>1 \), so it should be an increasing function (since base \( b>1 \) for \( \log_b x \) means increasing). Also, the domain is \( x>0 \), vertical asymptote at \( x = 0 \), and it passes through \( (1,0) \) (since \( \log_b 1=0 \)) and \( (7,1) \) (since \( \log_7 7 = 1 \)).

Step2: Analyze the graphs

• For a logarithmic function with \( b>1 \), as \( x \) increases, \( y \) increases. So we eliminate graphs that are decreasing (like if any have a curve going down as \( x \) increases).
• Check the key points: \( (1,0) \) and \( (7,1) \). The graph should pass through \( (1,0) \) (where \( x=1 \), \( y=0 \)) and rise slowly.
• Looking at the options, the correct graph should be increasing, have vertical asymptote at \( x=0 \), pass through \( (1,0) \), and for \( x = 7 \), \( y = 1 \). Typically, among the given options (from the description, assuming the standard log graph shape for base \( >1 \)), the graph that is increasing, has the vertical asymptote at \( x=0 \), and passes through \( (1,0) \) with the correct increasing trend. (Assuming the graph labeled appropriately, but from the options, the correct one is the one that matches the increasing log curve with base \( 7 \). If we consider typical log graphs, the correct graph would be the one that is increasing, so likely the one with the curve going up as \( x \) moves from right of 0 to larger values, passing through \( (1,0) \) and \( (7,1) \).)

Answer:

(Assuming the correct graph is the one that is increasing, has vertical asymptote at \( x = 0 \), passes through \( (1,0) \), so for the given options, if we assume the labels, but based on the standard, the correct graph is the one with the increasing logarithmic curve. If the options are A, B, C, D, and the correct one is the one that matches the properties, say if A is the correct increasing log graph, then the answer would be A. But based on typical log graphs for \( y=\log_7 x \), the correct graph is the one that is increasing, so the answer is the graph (option) that shows an increasing curve with vertical asymptote at \( x=0 \), passing through \( (1,0) \).)

(Note: Since the actual graphs are not fully visible, but based on the problem, the correct graph for \( y = \log_7 x \) (base \( >1 \)) is an increasing function, so the answer is the option with the increasing logarithmic curve, vertical asymptote at \( x=0 \), passing through \( (1,0) \). If we have to choose from A, B, C, D, and assuming the standard, the correct one is the one that is increasing, so for example, if A is the increasing log graph, then A. But without the exact graph details, but based on the function properties, the correct graph is the one that is increasing, has vertical asymptote at \( x=0 \), and passes through \( (1,0) \).)