To determine for which values of [tex]\( b \)[/tex] the function [tex]\( F(x) = \log_b x \)[/tex] is an increasing function, we must understand the properties of logarithmic functions:
1. Increasing vs Decreasing: The behavior of the logarithmic function [tex]\( \log_b x \)[/tex] depends on the base [tex]\( b \)[/tex].
- If [tex]\( b > 1 \)[/tex], the function [tex]\( \log_b x \)[/tex] is increasing.
- If [tex]\( 0 < b < 1 \)[/tex], the function [tex]\( \log_b x \)[/tex] is decreasing.
2. Reasoning:
- When [tex]\( b > 1 \)[/tex], as [tex]\( x \)[/tex] increases, the value of [tex]\( \log_b x \)[/tex] also increases, hence the function is increasing.
- When [tex]\( 0 < b < 1 \)[/tex], as [tex]\( x \)[/tex] increases, the value of [tex]\( \log_b x \)[/tex] decreases, hence the function is not increasing (it is decreasing).
3. Analyzing the options:
- Option A: [tex]\( b < 0 \)[/tex]: This is not valid since logarithmic functions are defined for positive bases only.
- Option B: [tex]\( b > 0 \)[/tex]: This is partially correct but too broad. It includes [tex]\( 0 < b < 1 \)[/tex] where the function decreases.
- Option C: [tex]\( b > 1 \)[/tex]: This is the correct condition. For [tex]\( b > 1 \)[/tex], [tex]\( \log_b x \)[/tex] is increasing.
- Option D: [tex]\( b < 1 \)[/tex]: This would include both negative values and the interval [tex]\( 0 < b < 1 \)[/tex]. It is not correct since an increasing function needs [tex]\( b > 1 \)[/tex].
Thus, the correct choice is:
C. [tex]\( b > 1 \)[/tex]
For the logarithmic function [tex]\( F(x) = \log_b x \)[/tex] to be an increasing function, [tex]\( b \)[/tex] must be greater than 1.
