Which Answer Represents The Range Of The Logarithmic Function Given Below? F ( X ) = Log ⁡ 0.5 X F(x)=\log_{0.5} X F ( X ) = Lo G 0.5 ​ X A. Y ≥ 0 Y \geq 0 Y ≥ 0 B. 1 \textless 0 1 \ \textless \ 0 1 \textless 0 C. Y \textgreater 0 Y \ \textgreater \ 0 Y \textgreater 0 D. All Real Numbers

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**Which Answer Represents the Range of the Logarithmic Function Given Below?**

Understanding Logarithmic Functions

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is used to solve equations that involve exponential expressions. The general form of a logarithmic function is:

f(x) = loga(x)

where a is the base of the logarithm and x is the input value.

The Given Logarithmic Function

The given logarithmic function is:

F(x) = log0.5(x)

This function has a base of 0.5, which is less than 1. This is an important characteristic of the function, as it affects the range of the function.

Understanding the Range of a Logarithmic Function

The range of a logarithmic function is the set of all possible output values. In other words, it is the set of all possible values that the function can take.

For a logarithmic function with a base of a, the range is all real numbers. However, when the base is less than 1, the range is restricted to non-negative real numbers.

Analyzing the Given Function

The given function F(x) = log0.5(x) has a base of 0.5, which is less than 1. Therefore, the range of this function is restricted to non-negative real numbers.

Evaluating the Answer Choices

Now, let's evaluate the answer choices:

A. y ≥ 0 B. 1 < 0 C. y > 0 D. All real numbers

Based on our analysis, the correct answer is:

A. y ≥ 0

This is because the range of the function F(x) = log0.5(x) is all non-negative real numbers.

Q&A

Q: What is the range of a logarithmic function with a base of a?

A: The range of a logarithmic function with a base of a is all real numbers.

Q: What happens to the range of a logarithmic function when the base is less than 1?

A: When the base is less than 1, the range of the function is restricted to non-negative real numbers.

Q: What is the range of the given function F(x) = log0.5(x)?

A: The range of the given function F(x) = log0.5(x) is all non-negative real numbers.

Q: Which answer choice represents the range of the given function?

A: The correct answer choice is A. y ≥ 0.

Q: Why is answer choice B incorrect?

A: Answer choice B is incorrect because 1 < 0 is not a valid statement. The correct statement is that the range of the function is restricted to non-negative real numbers.

Q: Why is answer choice C incorrect?

A: Answer choice C is incorrect because the range of the function is all non-negative real numbers, not just positive real numbers.

Q: Why is answer choice D incorrect?

A: Answer choice D is incorrect because the range of the function is restricted to non-negative real numbers, not all real numbers.

Conclusion

In conclusion, the range of the logarithmic function F(x) = log0.5(x) is all non-negative real numbers. The correct answer choice is A. y ≥ 0.