What are the domain and range of [tex]$f(x) = \log x - 5$[/tex]?

A. Domain: [tex]x \ \textgreater \ 0[/tex]; Range: all real numbers
B. Domain: [tex]x \ \textless \ 0[/tex]; Range: all real numbers
C. Domain: [tex]x \ \textgreater \ 5[/tex]; Range: [tex]y \ \textgreater \ 5[/tex]
D. Domain: [tex]x \ \textgreater \ 5[/tex]; Range: [tex]y \ \textgreater \ -5[/tex]



Answer :

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]