To determine the domain of the function
[tex]\[ f(x) = \log_7 (x - 3) - 5, \][/tex]
we need to consider when the expression inside the logarithm is defined. The logarithm function [tex]\( \log_b(y) \)[/tex] is defined only if [tex]\( y > 0 \)[/tex].
In this case, the argument of the logarithm [tex]\( (x - 3) \)[/tex] needs to be greater than 0. Thus, we set up the inequality:
[tex]\[ x - 3 > 0. \][/tex]
Now, we solve this inequality for [tex]\( x \)[/tex]:
[tex]\[ x > 3. \][/tex]
This means that [tex]\( x \)[/tex] must be greater than 3 for the function to be valid.
In interval notation, the domain of the function can be written as:
[tex]\[ (3, \infty). \][/tex]
So, the domain of the function [tex]\( f(x) = \log_7 (x - 3) - 5 \)[/tex] is
[tex]\[ (3, \infty). \][/tex]
