Answered

What is the range of y = logâ‚‚(x - 6)?


Aall real numbers greater than 6
Ball real numbers less than 6
Call real numbers not equal to 0
Dall real numbers

Verified Answer :

  • The domain of the function \(y = \log_2(x - 6)\) is \(x > 6\), as the logarithm argument must be positive.
  • Let \(t = x - 6\). Since \(x \in (6, \infty)\), \(t\) spans \((0, \infty)\).
  • The function \(y = \log_2(t)\) is continuous and strictly increasing, with limits of \(-\infty\) as \(t \to 0^+\) and \(\infty\) as \(t \to \infty\).
  • The range is the set of all real numbers, which matches option (a). \(\boxed{a}\)

Explanation

  1. Determine the domain of the function

To determine the range of the function \(y = \log_2(x - 6)\), we first identify the domain of the function. The logarithm function \(f(t) = \log_b(t)\) is only defined when the argument \(t\) is positive. In this case, the argument is \(x - 6\), so we must have the condition \(x - 6 > 0\), which simplifies to \(x > 6\). This means the domain of the function is the set of all real numbers strictly greater than 6, or in interval notation, \((6, \infty)\).

  1. Analyze the behavior of the function

Now consider the expression \(t = x - 6\). As \(x\) ranges over its domain \((6, \infty)\), the expression \(t\) takes on all values in the interval \((0, \infty)\). Consequently, the function \(y = \log_2(x - 6)\) becomes equivalent to finding the range of \(f(t) = \log_2(t)\) for \(t \in (0, \infty)\). The base of our logarithm is \(2\), which is greater than 1, implying that the function is strictly increasing on its domain.

  1. Evaluate the range of the function

As \(t\) approaches 0 from the positive side (\(t \to 0^+\)), the value of \(\log_2(t)\) approaches \(-\infty\). As \(t\) approaches positive infinity (\(t \to \infty\)), the value of \(\log_2(t)\) approaches \(\infty\). Since the function is continuous, by the Intermediate Value Theorem, the range of the function \(y = \log_2(t)\) for \(t \in (0, \infty)\) is the set of all real numbers, \((-\infty, \infty)\).

  1. Conclusion

Combining the information from the previous steps, we conclude that as \(x\) varies through its domain \((6, \infty)\), the function \(y = \log_2(x - 6)\) covers every possible real value. Therefore, the range of the function is all real numbers. Looking at the given options, this corresponds to option a.

Examples

The logarithmic function is fundamental in science and engineering. For example, it is used to measure sound intensity in decibels or earthquake magnitude on the Richter scale. These scales are designed to represent vast ranges of values in a more manageable logarithmic form. Understanding that logarithmic functions map positive inputs to all real numbers is key to interpreting these measurements correctly in real-world contexts, where the scale might need to extend across many orders of magnitude.