What is the range of $y=\log _2(x-6)$?
To determine the range of the function \(y = \log_2(x-6)\), we first consider its domain. The logarithm function \(\log_b(A)\) is defined only for positive arguments \(A > 0\). Thus, for the given function, we must have \(x - 6 > 0\), which simplifies to \(x > 6\). This means the domain of the function is the interval \((6, \infty)\).
Next, we examine the behavior of the function as \(x\) varies within its domain \((6, \infty)\). As the argument \((x-6)\) approaches 0 from the positive side (i.e., as \(x \to 6^+\)), \(\log_2(x-6)\) approaches \(-\infty\). Conversely, as the argument \((x-6)\) approaches positive infinity (i.e., as \(x \to \infty\)), \(\log_2(x-6)\) also approaches positive infinity.
Because the logarithmic function \(y = \log_2(x-6)\) is a continuous function on its entire domain \((6, \infty)\), the Intermediate Value Theorem ensures that it takes on every value between its limits. Since the function spans from \(-\infty\) to \(+\infty\), the range of the function is the set of all real numbers. This corresponds to the option 'all real numbers'.
Logarithmic functions are used in various fields, such as measuring the intensity of sounds (decibels) or the magnitude of earthquakes (Richter scale). For instance, if a sound intensity level is modeled by a logarithmic function, the range of possible decibel values reflects the wide span of detectable sound intensities by the human ear. Understanding the range of such functions allows scientists and engineers to predict and interpret these natural phenomena across a vast range of scales.