What is the range of $y=\log _2(x-6)$ ?
To determine the range of the function \(y = \log_2(x-6)\), we must first identify its domain. The logarithm function \(\log_b(t)\) is defined only for strictly positive arguments, so we require \(x-6 > 0\). Solving this inequality gives \(x > 6\). Thus, the domain of the function is the interval \((6, \infty)\).
Now, we consider the behavior of the function \(f(t) = \log_2(t)\) as \(t\) varies across its domain \((0, \infty)\). Because the base of the logarithm is \(2\), which is greater than \(1\), the function \(f(t) = \log_2(t)\) is continuous and strictly increasing. As \(t\) approaches \(0\) from the right (\(t \to 0^+\)), the function values approach \(-\infty\). Conversely, as \(t\) increases toward positive infinity (\(t \to \infty\)), the function values increase toward positive infinity (\(\infty\)).
Since the argument \(t = x-6\) takes on all values in \((0, \infty)\) as \(x\) varies in \((6, \infty)\), the function \(y = \log_2(x-6)\) also takes on all values that \(\log_2(t)\) covers on the interval \((0, \infty)\). Given that \(\log_2(t)\) approaches \(-\infty\) and \(+\infty\) at the boundaries, the function \(y = \log_2(x-6)\) covers every real number. Therefore, the range is the set of all real numbers.
Logarithmic functions are widely used in various fields, such as in acoustics for measuring sound intensity levels in decibels, or in chemistry for calculating the pH of solutions. For example, the pH of a solution is defined as \(pH = -\log_{10}[H^+]\), where \([H^+]\) is the hydrogen ion concentration. Understanding the range and domain of these functions is crucial for interpreting experimental data, where inputs like concentration are strictly positive but can result in a wide range of output values across the real number system.