Let's analyze the function [tex]\( g(x) = \log_5(x - 3) \)[/tex] to determine the x-intercept and the asymptote.
### Step-by-Step Solution:
1. Understanding the Function:
The given function is [tex]\( g(x) = \log_5(x - 3) \)[/tex]. This is a logarithmic function with the base 5 and the argument [tex]\(x - 3\)[/tex].
2. Finding the Vertical Asymptote:
For a logarithmic function of the form [tex]\( g(x) = \log_b(x - h) \)[/tex], the vertical asymptote occurs where the argument of the logarithm is 0 because the logarithm of 0 is undefined.
Set the argument equal to 0:
[tex]\[
x - 3 = 0 \implies x = 3
\][/tex]
Hence, the vertical asymptote is at [tex]\( x = 3 \)[/tex].
3. Finding the x-intercept:
The x-intercept of a function is the value of [tex]\( x \)[/tex] where [tex]\( g(x) = 0 \)[/tex].
Set the function equal to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[
\log_5(x - 3) = 0
\][/tex]
Recall that [tex]\( \log_b(1) = 0 \)[/tex] for any base [tex]\( b \)[/tex]. Thus:
[tex]\[
x - 3 = 1 \implies x = 4
\][/tex]
Hence, the x-intercept is at [tex]\( x = 4 \)[/tex].
4. Conclusion:
Based on our analysis, the x-intercept of the function [tex]\( g(x) = \log_5(x - 3) \)[/tex] is 4, and the asymptote is located at [tex]\( x = 3 \)[/tex].
### Choice Identification:
Given the choices, the correct statement is:
- The [tex]\(x\)[/tex]-intercept is 4, and the asymptote is located at [tex]\(x = 3\)[/tex].
