To find the relationship between the real zeros and the [tex]\( x \)[/tex]-intercepts of the function [tex]\( y = \log_4(x - 2) \)[/tex], let's follow these steps:
1. Set the function equal to zero to find [tex]\( x \)[/tex]-intercepts:
[tex]\[ y = \log_4(x - 2) = 0 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
The equation [tex]\( \log_4(x - 2) = 0 \)[/tex] can be rewritten in exponential form as:
[tex]\[ 4^0 = x - 2 \][/tex]
Since [tex]\( 4^0 = 1 \)[/tex]:
[tex]\[ 1 = x - 2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 1 + 2 \][/tex]
[tex]\[ x = 3 \][/tex]
So, the solution to the equation [tex]\( \log_4(x - 2) = 0 \)[/tex] is [tex]\( x = 3 \)[/tex]. Therefore, the function has an [tex]\( x \)[/tex]-intercept at [tex]\( x = 3 \)[/tex].
3. Check for vertical asymptotes:
The function [tex]\( y = \log_4(x - 2) \)[/tex] has a vertical asymptote where the argument of the logarithm is zero. The argument is [tex]\( x - 2 \)[/tex], so we set it to zero:
[tex]\[ x - 2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 2 \][/tex]
Thus, the function has a vertical asymptote at [tex]\( x = 2 \)[/tex].
From this analysis, we can conclude:
- The function [tex]\( y = \log_4(x - 2) \)[/tex] has an [tex]\( x \)[/tex]-intercept at [tex]\( x = 3 \)[/tex].
- There is a vertical asymptote at [tex]\( x = 2 \)[/tex].
Given the choices, the correct relationship is:
"When you set the function equal to zero, the solution is [tex]\( x = 3 \)[/tex]; therefore, the graph has an [tex]\( x \)[/tex]-intercept at [tex]\( x = 3 \)[/tex]."
