Question

for what values of b will ( f(x) = log_b x ) be an increasing function?
a. ( b < 1 )
b. ( b < 0 )
c. ( b > 0 )
d. ( 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 \( b
eq1 \) (domain of logarithm base). For the function to be increasing, we use the property of logarithmic functions: when \( b > 1 \), the logarithmic function \( y = \log_{b}x \) is an increasing function (as \( x \) increases, \( \log_{b}x \) increases); when \( 0 < b < 1 \), the function is decreasing.

Step2: Evaluate Each Option

• Option A: \( b < 1 \). But if \( 0 < b < 1 \), the function is decreasing, so A is incorrect.
• Option B: \( b < 0 \). The base of a logarithm cannot be negative (since \( \log_{b}x=\frac{\ln x}{\ln b} \), and \( \ln b \) is undefined for \( b\leq0 \)), so B is invalid.
• Option C: \( b > 0 \). But \( b \) also needs to be greater than 1 for increasing (since \( 0 < b < 1 \) gives a decreasing function), so C is incorrect.
• Option D: \( b > 1 \). This satisfies the condition for \( F(x)=\log_{b}x \) to be an increasing function.

Answer:

D. \( b > 1 \)