To determine the interval over which the function [tex]\( f(x) = \log_4(x - 2) \)[/tex] is positive, let's analyze this step by step.
1. Understanding the function:
The function [tex]\( f(x) = \log_4(x - 2) \)[/tex] involves a logarithm with base 4. A logarithmic function [tex]\( \log_b(y) \)[/tex] is positive if [tex]\( y > 1 \)[/tex], zero if [tex]\( y = 1 \)[/tex], and negative if [tex]\( 0 < y < 1 \)[/tex].
2. Setting up the inequality:
We want to find the values of [tex]\( x \)[/tex] such that [tex]\( \log_4(x - 2) > 0 \)[/tex].
3. Interpreting the logarithm inequality:
For the logarithm [tex]\( \log_4(x - 2) \)[/tex] to be positive, the argument [tex]\( (x - 2) \)[/tex] must be greater than 1. This is because:
[tex]\[
\log_4(y) > 0 \quad \text{if and only if} \quad y > 1
\][/tex]
Therefore:
[tex]\[
x - 2 > 1
\][/tex]
4. Solving the inequality:
Solving [tex]\( x - 2 > 1 \)[/tex],
[tex]\[
x > 3
\][/tex]
5. Conclusion:
The function [tex]\( f(x) \)[/tex] is positive for [tex]\( x > 3 \)[/tex]. In interval notation, this is:
[tex]\[
(3, \infty)
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{\text{A. } (3, \infty)}
\][/tex]
