Question

for what values of b will f(x) = \log_b x be a decreasing function?

a. 0 < b < 1

b. b < 0

c. b > 0

d. 0 > b > -1

Explanation:

Step1: Recall Logarithmic Function Properties

The general form of a logarithmic function is \( F(x)=\log_b x \), where the base \( b \) must satisfy \( b > 0 \) and \( b
eq1 \) (since the logarithm is only defined for positive bases not equal to 1).

Step2: Analyze Monotonicity of Logarithmic Functions

For the logarithmic function \( y = \log_b x \):

• If \( b>1 \), the function is increasing (as \( x \) increases, \( y \) increases).
• If \( 0 < b < 1 \), the function is decreasing (as \( x \) increases, \( y \) decreases).

Now, let's analyze the options:

• Option A: \( 0 < b < 1 \) satisfies the condition for \( \log_b x \) to be decreasing (and also the domain of the base for a logarithm).
• Option B: \( b < 0 \) is invalid because the base of a logarithm must be positive (and not equal to 1).
• Option C: \( b>0 \) includes cases where \( b > 1 \) (in which the function is increasing) and \( 0 < b < 1 \) (decreasing), so it's not specific to decreasing.
• Option D: \( 0 > b > - 1 \) (i.e., \( - 1 < b < 0 \)) is invalid because the base of a logarithm must be positive.

Answer:

A. \( 0 < b < 1 \)