QuestionIf we graph of a ioganthernic function is given. Seslect the function for the graph from the oppliogrim A. $f(x)=\log _{4}(x-2)$ B. $f(x)=\log _{4} x$ C. $f(x)=\log _{4}(x+2)$ D. $f(x)=\log _{4} x+2$

Studdy Solution

STEP 1

1. The logarithmic function has a vertical asymptote at $x = 2$ .
2. The function passes through the point $(3, 0)$ .
3. The base of the logarithm is $4$ .
4. We need to match the graph to one of the provided function options.

STEP 2

1. Analyze the vertical asymptote to understand the shift in the function.
2. Verify which function passes through the point $(3, 0)$ .
3. Conclude the correct function based on the conditions.

STEP 3

Analyze the vertical asymptote. For a logarithmic function $f(x) = \log_{b}(x - h)$ , the vertical asymptote occurs at $x = h$ .
Given the vertical asymptote at $x = 2$ , we need a function of the form $f(x) = \log_{4}(x - 2)$ .

STEP 4

Verify which function passes through the point $(3, 0)$ .
For $f(x) = \log_{4}(x - 2)$ :
$f(3) = \log_{4}(3 - 2) = \log_{4}(1) = 0$
This function passes through the point $(3, 0)$ .

STEP 5

Conclude that the correct function is $f(x) = \log_{4}(x - 2)$ .
The function with a vertical asymptote at $x = 2$ and passing through $(3, 0)$ is:
$\boxed{f(x) = \log_{4}(x - 2)}$

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