- The domain of the logarithmic function \(f(x) = \log x - 5\) is determined by the requirement that the argument \(x\) must be positive, giving \(x > 0\).
- The range of the logarithmic function \(\log x\) covers all real numbers, and a vertical shift of 5 units does not change this range.
- Thus, the domain is \(x > 0\) and the range is all real numbers.
- The correct option is: \(\boxed{\text{domain: } x > 0; \text{range: all real numbers}}\).
Explanation
- Analyzing the Domain
To determine the domain and range of the function \(f(x) = \log x - 5\), we first examine the base function \(g(x) = \log x\). The domain of a logarithmic function, where the base is implicitly 10, is defined by the requirement that the argument must be strictly positive. In this case, the argument is \(x\), so we must have \(x > 0\). This condition defines the set of all possible input values for the function.
- Analyzing the Range
Next, we consider the range of the function \(g(x) = \log x\). The logarithmic function is defined for all positive real numbers and its values cover all real numbers from \(-\infty\) to \(\infty\). Therefore, the range of \(g(x) = \log x\) is \((-\infty, \infty)\), which represents all real numbers. When we apply the transformation \(f(x) = g(x) - 5\), we are shifting the graph of \(\log x\) vertically downward by 5 units. A vertical translation does not change the set of output values when the original range is all real numbers. Thus, the range of \(f(x) = \log x - 5\) remains the set of all real numbers.
- Comparing with Options and Conclusion
Combining our findings, the domain is \(x > 0\) and the range is all real numbers. Comparing this to the provided options, it matches the first choice: domain: \(x > 0\); range: all real numbers.
Examples
Logarithmic functions are essential in many scientific applications. For example, the pH of a solution is defined by the negative logarithm of the concentration of hydrogen ions: \(\text{pH} = -\log[H^+]\). Just like our function, the concentration of hydrogen ions \([H^+]\) must be positive, which mirrors the domain requirement of the logarithm function. Understanding these properties helps scientists calculate acidity levels and interpret chemical data accurately.