Answer :
To determine the domain and range of the logarithmic function [tex]\( F(x) = \log_4 x \)[/tex], let's analyze the properties of logarithmic functions.
### Domain:
The domain of any logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b > 1 \)[/tex]) is the set of all positive real numbers. This implies that [tex]\( x \)[/tex] must be greater than 0 because the logarithm of a non-positive number is undefined.
### Range:
The range of a logarithmic function is the set of all real numbers. This means that for any real number [tex]\( y \)[/tex], there exists some [tex]\( x > 0 \)[/tex] such that [tex]\( \log_b(x) = y \)[/tex].
Let's consider the options:
A. The domain is [tex]\( x < 0 \)[/tex], and the range is all real numbers.
- Incorrect: The domain [tex]\( x < 0 \)[/tex] is incorrect for a logarithmic function since [tex]\( x \)[/tex] must be positive.
B. The domain is [tex]\( x > 0 \)[/tex], and the range is [tex]\( y > 0 \)[/tex].
- Incorrect: While the domain is correct, the range [tex]\( y > 0 \)[/tex] is incorrect. The range of a logarithmic function includes all real numbers, not just positive numbers.
C. The domain is [tex]\( x < 0 \)[/tex], and the range is [tex]\( y < 0 \)[/tex].
- Incorrect: Both the domain and range are incorrect. The domain should be [tex]\( x > 0 \)[/tex] and the range should include all real numbers.
D. The domain is [tex]\( x > 0 \)[/tex], and the range is all real numbers.
- Correct: This option correctly states that the domain is [tex]\( x > 0 \)[/tex] and the range encompasses all real numbers.
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
### Domain:
The domain of any logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b > 1 \)[/tex]) is the set of all positive real numbers. This implies that [tex]\( x \)[/tex] must be greater than 0 because the logarithm of a non-positive number is undefined.
### Range:
The range of a logarithmic function is the set of all real numbers. This means that for any real number [tex]\( y \)[/tex], there exists some [tex]\( x > 0 \)[/tex] such that [tex]\( \log_b(x) = y \)[/tex].
Let's consider the options:
A. The domain is [tex]\( x < 0 \)[/tex], and the range is all real numbers.
- Incorrect: The domain [tex]\( x < 0 \)[/tex] is incorrect for a logarithmic function since [tex]\( x \)[/tex] must be positive.
B. The domain is [tex]\( x > 0 \)[/tex], and the range is [tex]\( y > 0 \)[/tex].
- Incorrect: While the domain is correct, the range [tex]\( y > 0 \)[/tex] is incorrect. The range of a logarithmic function includes all real numbers, not just positive numbers.
C. The domain is [tex]\( x < 0 \)[/tex], and the range is [tex]\( y < 0 \)[/tex].
- Incorrect: Both the domain and range are incorrect. The domain should be [tex]\( x > 0 \)[/tex] and the range should include all real numbers.
D. The domain is [tex]\( x > 0 \)[/tex], and the range is all real numbers.
- Correct: This option correctly states that the domain is [tex]\( x > 0 \)[/tex] and the range encompasses all real numbers.
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
