To determine the domain and range of the function [tex]\( f(x) = \log x - 5 \)[/tex], let's analyze it in detail.
### Domain:
1. The logarithmic function [tex]\( \log x \)[/tex] is defined only for [tex]\( x > 0 \)[/tex]. This is because you cannot take the logarithm of a non-positive number (zero or negative).
2. Since [tex]\( f(x) = \log x - 5 \)[/tex] includes the logarithmic function, the domain of [tex]\( f(x) \)[/tex] remains the same: [tex]\( x > 0 \)[/tex].
Therefore, the domain of [tex]\( f(x) = \log x - 5 \)[/tex] is [tex]\( x > 0 \)[/tex].
### Range:
1. The logarithmic function [tex]\( \log x \)[/tex] can take any real number as its output. As [tex]\( x \)[/tex] increases, [tex]\( \log x \)[/tex] increases without bound. Similarly, as [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( \log x \)[/tex] decreases without bound.
2. Subtracting 5 from [tex]\( \log x \)[/tex] shifts the entire log function down by 5 units. However, this shifting does not change the fact that the function can still take any real value. Specifically, for any real number [tex]\( y \)[/tex], we can find some [tex]\( x \)[/tex] such that [tex]\( \log x = y + 5 \)[/tex].
Thus, the range of [tex]\( f(x) = \log x - 5 \)[/tex] is all real numbers.
Based on this analysis:
- The domain of [tex]\( f(x) = \log x - 5 \)[/tex] is [tex]\( x > 0 \)[/tex].
- The range of [tex]\( f(x) = \log x - 5 \)[/tex] is all real numbers.
Among the given options, the one that matches this is:
- Domain: [tex]\( x > 0 \)[/tex]; Range: all real numbers.
Therefore, the correct option is:
[tex]\[ \text{domain: } x > 0; \text{ range: } \text{all real numbers} \][/tex]
