To determine the correctness of the statement, "The domain of [tex]\( F(x) = \log_b x \)[/tex] is the set of all real numbers," let's analyze the domain of the logarithmic function.
1. Definition of the Logarithmic Function: The logarithmic function [tex]\( F(x) = \log_b x \)[/tex] is defined for a positive base [tex]\( b \)[/tex] (where [tex]\( b > 0 \)[/tex] and [tex]\( b \neq 1 \)[/tex]).
2. Domain Analysis: For the function [tex]\( \log_b x \)[/tex] to be defined, the argument [tex]\( x \)[/tex] must be positive. This means [tex]\( x \)[/tex] must be greater than 0.
Mathematically, the domain of [tex]\( \log_b x \)[/tex] is:
[tex]\[
x > 0
\][/tex]
3. Examining the Statement: The statement claims that the domain is the set of all real numbers. However, from our analysis, we see that the domain is actually the set of all positive real numbers, [tex]\( x > 0 \)[/tex], and not all real numbers.
4. Conclusion: The domain of [tex]\( F(x) = \log_b x \)[/tex] excludes zero and negative numbers, thus it is not the set of all real numbers. Therefore, the statement is not correct.
The statement, "The domain of [tex]\( F(x) = \log_b x \)[/tex] is the set of all real numbers," is:
B. False
