Comparing the Possible Answers

Oct 13, 2025 by windailys.com 18 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their domain and range is crucial for solving various mathematical problems. In this article, we will explore the range of the function $$y=\log_2(x-6)$$ and discuss the possible answers.

What is the Domain of a Logarithmic Function?

Before we can determine the range of a logarithmic function, we need to understand its domain. The domain of a logarithmic function is the set of all possible input values (x) for which the function is defined. In the case of the function $$y=\log_2(x-6)$$, the domain is all real numbers greater than 6, since the logarithm of a non-positive number is undefined.

Why is the Domain Important?

The domain of a logarithmic function is important because it determines the possible output values (y) of the function. If the input value (x) is not within the domain, the function is undefined, and the output value (y) is also undefined.

What is the Range of a Logarithmic Function?

Now that we have discussed the domain of the function $$y=\log_2(x-6)$$, we can determine its range. The range of a logarithmic function is the set of all possible output values (y) for which the function is defined.

The Range of the Function $$y=\log_2(x-6)$$

To determine the range of the function $$y=\log_2(x-6)$$, we need to consider the possible output values (y) for all real numbers greater than 6. Since the logarithm of a number is always positive, the range of the function is all real numbers greater than 0.

Why is the Range Important?

The range of a logarithmic function is important because it determines the possible output values (y) of the function. If the output value (y) is not within the range, the function is undefined, and the input value (x) is also undefined.

Comparing the Possible Answers

Now that we have determined the range of the function $$y=\log_2(x-6)$$, we can compare it with the possible answers.

A. All real numbers not equal to 0: This is not correct, since the range of the function is all real numbers greater than 0.
B. All real numbers less than 6: This is not correct, since the domain of the function is all real numbers greater than 6.
C. All real numbers greater than 6: This is not correct, since the range of the function is all real numbers greater than 0.
D. All real numbers: This is not correct, since the range of the function is all real numbers greater than 0.

Conclusion

In conclusion, the range of the function $$y=\log_2(x-6)$$ is all real numbers greater than 0. This is because the logarithm of a number is always positive, and the domain of the function is all real numbers greater than 6.

Final Answer

Introduction

In our previous article, we discussed the domain and range of the function $$y=\log_2(x-6)$$. We determined that the domain is all real numbers greater than 6 and the range is all real numbers greater than 0. In this article, we will answer some frequently asked questions about the domain and range of logarithmic functions.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all possible input values (x) for which the function is defined. In the case of the function $$y=\log_2(x-6)$$, the domain is all real numbers greater than 6, since the logarithm of a non-positive number is undefined.

Q: Why is the domain important?

A: The domain of a logarithmic function is important because it determines the possible output values (y) of the function. If the input value (x) is not within the domain, the function is undefined, and the output value (y) is also undefined.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is the set of all possible output values (y) for which the function is defined. In the case of the function $$y=\log_2(x-6)$$, the range is all real numbers greater than 0.

Q: Why is the range important?

A: The range of a logarithmic function is important because it determines the possible output values (y) of the function. If the output value (y) is not within the range, the function is undefined, and the input value (x) is also undefined.

Q: How do I determine the domain and range of a logarithmic function?

A: To determine the domain and range of a logarithmic function, you need to consider the following:

The domain is all real numbers greater than the base of the logarithm (in this case, 2).
The range is all real numbers greater than 0.

Q: What is the difference between the domain and range of a logarithmic function?

A: The domain of a logarithmic function is the set of all possible input values (x) for which the function is defined, while the range is the set of all possible output values (y) for which the function is defined.

Q: Can I have a negative input value for a logarithmic function?

A: No, you cannot have a negative input value for a logarithmic function. The logarithm of a non-positive number is undefined.

Q: Can I have a zero input value for a logarithmic function?

A: No, you cannot have a zero input value for a logarithmic function. The logarithm of zero is undefined.

Q: Can I have a fractional input value for a logarithmic function?

A: Yes, you can have a fractional input value for a logarithmic function. However, the input value must be greater than the base of the logarithm (in this case, 2).

Q: Can I have a large input value for a logarithmic function?

A: Yes, you can have a large input value for a logarithmic function. However, the input value must be greater than the base of the logarithm (in this case, 2).

Conclusion

In conclusion, understanding the domain and range of logarithmic functions is crucial for solving various mathematical problems. By following the steps outlined in this article, you can determine the domain and range of a logarithmic function and use this knowledge to solve problems.

Final Answer

The final answer is: The domain of a logarithmic function is all real numbers greater than the base of the logarithm, and the range is all real numbers greater than 0.