Question

identify a possible base ( b ) of this logarithmic function.
(1 point)
( \bigcirc ) ( b = 0 )
( \bigcirc ) ( b = -4 )
( \bigcirc ) ( b = \frac{1}{4} )
( \bigcirc ) ( b = 4 )

Explanation:

Step1: Recall Logarithmic Function Properties

The general form of a logarithmic function is \( y = \log_b(x) \), where \( b>0 \), \( b
eq1 \). So we can eliminate options with \( b = 0 \) (since \( \log_0(x) \) is undefined) and \( b=- 4 \) (base must be positive). Now we have \( b=\frac{1}{4} \) and \( b = 4 \) left.

Step2: Analyze the Graph's Behavior

The graph of \( y=\log_b(x) \) is decreasing. For a logarithmic function \( y=\log_b(x) \), if \( 0 < b<1 \), the function is decreasing; if \( b > 1 \), the function is increasing. The given graph is decreasing, so \( 0 < b<1 \). Among the remaining options, \( b=\frac{1}{4} \) satisfies \( 0<\frac{1}{4}<1 \), while \( b = 4 \) is greater than 1 (would give an increasing graph, which doesn't match).

Answer:

\( b=\frac{1}{4} \) (corresponding to the option \( b=\frac{1}{4} \))