Find the domain of the function.
$g(x)=\ln (x-8)$
The function provided is \(g(x) = \ln(x - 8)\). To find its domain, we must determine the set of all possible input values \(x\) for which the function is well-defined. The natural logarithm function, denoted as \(\ln(u)\), is defined if and only if its argument \(u\) is strictly greater than zero, i.e., \(u > 0\).
Given that the function is \(g(x) = \ln(x - 8)\), we identify the argument as \(x - 8\). According to the definition of the natural logarithm, we must satisfy the inequality \(x - 8 > 0\). Adding 8 to both sides of the inequality gives us \(x > 8\).
The condition \(x > 8\) means that the function \(g(x) = \ln(x - 8)\) is defined for all real numbers \(x\) greater than 8. In interval notation, this domain is represented as \((8, \infty)\). Therefore, any value of \(x\) less than or equal to 8 would make the argument of the logarithm less than or equal to 0, which is outside the domain of the natural logarithm function. The domain of the function is \(\boxed{(8, \infty)}\).
Understanding the domain of logarithmic functions is crucial in fields like finance and acoustics. For instance, in modeling the decibel level of a sound based on its intensity relative to a threshold, the intensity ratio must be positive. By determining the domain, engineers ensure that their models are mathematically valid and avoid undefined calculations when analyzing real-world data like noise levels or investment growth rates.