What Is The Range Of $y = \log_2(x-6$\]?A. All Real Numbers Not Equal To 0 B. All Real Numbers Less Than 6 C. All Real Numbers Greater Than 6 D. All Real Numbers

Understanding the Basics of Logarithmic Functions

Logarithmic functions are a crucial part of mathematics, and they play a vital role in various fields, including science, engineering, and economics. A logarithmic function is the inverse of an exponential function, and it is defined as the power to which a base number must be raised to produce a given value. In this article, we will focus on the range of the logarithmic function $y = \log_2(x-6)$.

The Domain of the Logarithmic Function

Before we can determine the range of the function, we need to understand its domain. The domain of a function is the set of all possible input values for which the function is defined. In the case of the logarithmic function $y = \log_2(x-6)$, the base of the logarithm is 2, and the argument of the logarithm is $(x-6)$. Since the logarithm of a non-positive number is undefined, we must have $(x-6) > 0$, which implies that $x > 6$. Therefore, the domain of the function is all real numbers greater than 6.

The Range of the Logarithmic Function

Now that we have determined the domain of the function, we can focus on finding its range. The range of a function is the set of all possible output values for which the function is defined. In the case of the logarithmic function $y = \log_2(x-6)$, we can see that the output value $y$ is always positive, since the logarithm of a positive number is always positive. Moreover, as the input value $x$ increases, the output value $y$ also increases, but at a slower rate. This is because the logarithmic function is a non-linear function, and its rate of change decreases as the input value increases.

Analyzing the Options

Now that we have a good understanding of the logarithmic function $y = \log_2(x-6)$, we can analyze the options given in the problem.

• Option A: All real numbers not equal to 0. This option is incorrect, since the logarithmic function $y = \log_2(x-6)$ is always positive, and it is never equal to 0.
• Option B: All real numbers less than 6. This option is incorrect, since the domain of the function is all real numbers greater than 6.
• Option C: All real numbers greater than 6. This option is correct, since the domain of the function is all real numbers greater than 6, and the range of the function is all real numbers greater than 0.
• Option D: All real numbers. This option is incorrect, since the range of the function is not all real numbers, but rather all real numbers greater than 0.

Conclusion

In conclusion, the range of the logarithmic function $y = \log_2(x-6)$ is all real numbers greater than 0. This is because the domain of the function is all real numbers greater than 6, and the output value $y$ is always positive. Therefore, the correct answer is option C: All real numbers greater than 6.

Frequently Asked Questions

• What is the domain of the logarithmic function $y = \log_2(x-6)$? The domain of the function is all real numbers greater than 6.
• What is the range of the logarithmic function $y = \log_2(x-6)$? The range of the function is all real numbers greater than 0.
• Why is the logarithmic function $y = \log_2(x-6)$ always positive? The logarithmic function $y = \log_2(x-6)$ is always positive because the logarithm of a positive number is always positive.

Final Answer

The final answer is option C: All real numbers greater than 6.