To determine the domain of the logarithmic function [tex]\( F(x) = \log_9 x \)[/tex], we need to understand the properties of logarithmic functions.
A logarithmic function [tex]\( F(x) = \log_b(x) \)[/tex] is defined only for positive real numbers [tex]\( x \)[/tex]. This means [tex]\( x \)[/tex] must be greater than 0. Therefore, the domain of [tex]\( F(x) = \log_9 x \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x > 0 \)[/tex].
Let's review the given options:
- Option A: [tex]\( x \geq 0 \)[/tex] — This includes 0, but the logarithmic function is not defined at [tex]\( x = 0 \)[/tex].
- Option B: [tex]\( x < 0 \)[/tex] — This includes negative values, but the logarithmic function is not defined for negative [tex]\( x \)[/tex].
- Option C: [tex]\( x > 0 \)[/tex] — This correctly represents that [tex]\( x \)[/tex] must be greater than 0.
- Option D: All real numbers — This includes both positive and negative numbers and zero, but the logarithmic function is only defined for positive [tex]\( x \)[/tex].
The correct option is:
C. [tex]\( x > 0 \)[/tex]
