- The domain of the logarithmic function \(f(x) = \log(x+6) - 4\) is found by setting the argument \(x+6 > 0\), resulting in \(x > -6\).
- The range of any basic logarithmic function \(\log(u)\) is all real numbers \((-\infty, \infty)\).
- A vertical shift by \(-4\) does not change the range of the function.
- The final answer is: \(\boxed{\text{domain: } x > -6; \text{ range: all real numbers}}\).
Explanation
- Determining the Domain
The given function is \(f(x) = \log(x+6) - 4\). To determine the domain, we recall that the logarithmic function \(\log(u)\) is only defined when its argument \(u\) is strictly greater than zero. Therefore, we set the condition \(x + 6 > 0\). Solving this inequality, we find \(x > -6\). Thus, the domain of the function is the set of all real numbers \(x\) such that \(x > -6\).
- Determining the Range
The range of the basic logarithmic function \(\log(u)\) for \(u > 0\) is the set of all real numbers, which can be written as \((-\infty, \infty)\). The function \(f(x) = \log(x+6) - 4\) involves a horizontal shift (left by 6 units) and a vertical shift (down by 4 units). While a horizontal shift affects the domain, a vertical shift does not alter the fact that the output of the logarithmic function covers all possible real values. Subtracting 4 from the entire range \((-\infty, \infty)\) still results in the set of all real numbers. Therefore, the range is all real numbers.
- Conclusion
Based on our analysis, the domain is \(x > -6\) and the range is all real numbers. Comparing this to the given options, the correct choice is: domain: \(x > -6\); range: all real numbers.
Examples
Logarithmic functions are used extensively in fields like chemistry and acoustics. For instance, the pH level of a substance is defined as \(pH = -\log_{10}[H^+]\), where \([H^+]\) is the concentration of hydrogen ions. Knowing the domain of the logarithm is crucial, as you cannot have a negative concentration of ions. Similarly, sound intensity levels in decibels are measured using a logarithmic scale, showing how our senses perceive signals over a vast range of magnitudes. Mastering these functions helps in understanding real-world processes that occur on exponential or logarithmic scales.