Answer :
To determine the range of the logarithmic function [tex]\( F(x) = \log_{0.3} x \)[/tex], we need to analyze the properties of logarithmic functions with a base between 0 and 1.
Let’s start by recalling some properties of logarithmic functions:
1. Base Interval: If the base [tex]\( b \)[/tex] (where [tex]\( 0 < b < 1 \)[/tex]), the logarithmic function [tex]\( \log_b x \)[/tex] is a decreasing function.
2. Behavior around [tex]\( x = 1 \)[/tex]: The value of [tex]\( F(x) \)[/tex] when [tex]\( x = 1 \)[/tex] is [tex]\( \log_{0.3} 1 = 0 \)[/tex].
3. Domain: The domain of [tex]\( \log_{0.3} x \)[/tex] includes all positive real numbers, i.e., [tex]\( x > 0 \)[/tex].
### Step-by-Step Analysis
1. Decreasing Nature: Since the base is [tex]\( 0.3 \)[/tex] (which lies between 0 and 1), the function is decreasing. This means as [tex]\( x \)[/tex] increases, [tex]\( \log_{0.3} x \)[/tex] decreases.
2. Range Analysis for [tex]\( x > 1 \)[/tex]:
- When [tex]\( x > 1 \)[/tex], because the function is decreasing, [tex]\( \log_{0.3} x \)[/tex] will produce negative values since [tex]\( \log_{0.3} 1 = 0 \)[/tex] and the function must decrease as [tex]\( x \)[/tex] increases.
3. Range Analysis for [tex]\( 0 < x < 1 \)[/tex]:
- When [tex]\( 0 < x < 1 \)[/tex], the logarithm of numbers between 0 and 1 with a base less than 1 results in positive values. For example, [tex]\( \log_{0.3} 0.3 = 1 \)[/tex].
### Conclusion
By combining these analyses, since the function’s value can cover all real numbers:
- For [tex]\( x > 1 \)[/tex]: [tex]\( \log_{0.3} x \)[/tex] ranges from 0 to [tex]\(-\infty\)[/tex], covering negative values.
- For [tex]\( 0 < x < 1 \)[/tex]: [tex]\( \log_{0.3} x \)[/tex] ranges from 0 to [tex]\( +\infty\)[/tex], covering positive values.
Thus, the range of [tex]\( F(x) = \log_{0.3} x \)[/tex] is all real numbers. This encompasses both positive and negative values as well as zero.
Therefore, the correct answer is:
D. All real numbers
Let’s start by recalling some properties of logarithmic functions:
1. Base Interval: If the base [tex]\( b \)[/tex] (where [tex]\( 0 < b < 1 \)[/tex]), the logarithmic function [tex]\( \log_b x \)[/tex] is a decreasing function.
2. Behavior around [tex]\( x = 1 \)[/tex]: The value of [tex]\( F(x) \)[/tex] when [tex]\( x = 1 \)[/tex] is [tex]\( \log_{0.3} 1 = 0 \)[/tex].
3. Domain: The domain of [tex]\( \log_{0.3} x \)[/tex] includes all positive real numbers, i.e., [tex]\( x > 0 \)[/tex].
### Step-by-Step Analysis
1. Decreasing Nature: Since the base is [tex]\( 0.3 \)[/tex] (which lies between 0 and 1), the function is decreasing. This means as [tex]\( x \)[/tex] increases, [tex]\( \log_{0.3} x \)[/tex] decreases.
2. Range Analysis for [tex]\( x > 1 \)[/tex]:
- When [tex]\( x > 1 \)[/tex], because the function is decreasing, [tex]\( \log_{0.3} x \)[/tex] will produce negative values since [tex]\( \log_{0.3} 1 = 0 \)[/tex] and the function must decrease as [tex]\( x \)[/tex] increases.
3. Range Analysis for [tex]\( 0 < x < 1 \)[/tex]:
- When [tex]\( 0 < x < 1 \)[/tex], the logarithm of numbers between 0 and 1 with a base less than 1 results in positive values. For example, [tex]\( \log_{0.3} 0.3 = 1 \)[/tex].
### Conclusion
By combining these analyses, since the function’s value can cover all real numbers:
- For [tex]\( x > 1 \)[/tex]: [tex]\( \log_{0.3} x \)[/tex] ranges from 0 to [tex]\(-\infty\)[/tex], covering negative values.
- For [tex]\( 0 < x < 1 \)[/tex]: [tex]\( \log_{0.3} x \)[/tex] ranges from 0 to [tex]\( +\infty\)[/tex], covering positive values.
Thus, the range of [tex]\( F(x) = \log_{0.3} x \)[/tex] is all real numbers. This encompasses both positive and negative values as well as zero.
Therefore, the correct answer is:
D. All real numbers
