To determine the domain and range of the function [tex]\( f(x) = \log x - 5 \)[/tex], let's go through the problem step-by-step.
### Domain:
The domain of a function includes all the values of [tex]\( x \)[/tex] for which the function is defined.
1. We start with the component [tex]\( \log x \)[/tex].
2. The logarithm function, [tex]\( \log x \)[/tex], is defined only for [tex]\( x > 0 \)[/tex] because the logarithm of zero or a negative number is undefined in the realm of real numbers.
3. Therefore, [tex]\( \log x \)[/tex] exists only when [tex]\( x > 0 \)[/tex].
So, the domain of the function [tex]\( f(x) = \log x - 5 \)[/tex] is [tex]\( x > 0 \)[/tex].
### Range:
The range of a function consists of all possible values of [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) that the function can take.
1. Consider the behavior of [tex]\( \log x \)[/tex]. The logarithm function [tex]\( \log x \)[/tex] can take any real number because:
- As [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( \log x \)[/tex] approaches [tex]\(-\infty\)[/tex].
- As [tex]\( x \)[/tex] increases towards infinity, [tex]\( \log x \)[/tex] increases towards [tex]\( +\infty \)[/tex].
2. Subtracting 5 from [tex]\( \log x \)[/tex] shifts the entire graph of the logarithm function downward by 5 units. However, this transformation does not change the fact that [tex]\( \log x \)[/tex] can take any real value; it just shifts the range.
3. Consequently, [tex]\( \log x - 5 \)[/tex] can also take any real number.
So, the range of the function [tex]\( f(x) = \log x - 5 \)[/tex] is all real numbers.
### Summary:
The domain of the function [tex]\( f(x) = \log x - 5 \)[/tex] is [tex]\( x > 0 \)[/tex], and the range is all real numbers.
Therefore, the correct answer is:
- Domain: [tex]\( x > 0 \)[/tex]
- Range: all real numbers
