To determine the range of the function [tex]\( y = \log_2(x - 6) \)[/tex], let's analyze it step-by-step.
1. Understanding the Function:
- The function [tex]\( y = \log_2(x - 6) \)[/tex] is a logarithmic function with base 2.
- For the logarithmic function to be defined, the argument [tex]\( x - 6 \)[/tex] must be greater than 0.
2. Domain of the Function:
- Solving the inequality [tex]\( x - 6 > 0 \)[/tex] gives [tex]\( x > 6 \)[/tex].
- So, the domain of the function is all real numbers greater than 6.
3. Behavior of Logarithmic Functions:
- The logarithmic function [tex]\( \log_2(z) \)[/tex] (and logarithmic functions in general regardless of the base) is defined for [tex]\( z > 0 \)[/tex].
- The output of a logarithmic function, or its range, is all real numbers.
4. Applying to the Given Function:
- For [tex]\( y = \log_2(x - 6) \)[/tex], [tex]\( x - 6 \)[/tex] can take any positive value as [tex]\( x \)[/tex] varies over the domain.
- As [tex]\( x - 6 \)[/tex] varies over positive values, the function [tex]\( y = \log_2(x - 6) \)[/tex] can produce any real number.
5. Conclusion:
- The range of the function [tex]\( y = \log_2(x - 6) \)[/tex] is all real numbers.
Thus, the answer is:
[tex]\[ \text{all real numbers} \][/tex]
