Questions: Find the logarithmic function, whose graph is shown below: f(x)=log3(x-5) f(x)=log6(x-3) f(x)=log3(x+5) f(x)=log5(x+3) f(x)=log5(x-3) None of the above

Find the logarithmic function, whose graph is shown below: f(x)=log3(x-5) f(x)=log6(x-3) f(x)=log3(x+5) f(x)=log5(x+3) f(x)=log5(x-3) None of the above
Transcript text: Find the logarithmic function, whose graph is shown below: \[ f(x)=\log _{3}(x-5) \] \[ f(x)=\log _{6}(x-3) \] \[ f(x)=\log _{3}(x+5) \] \[ f(x)=\log _{5}(x+3) \] \[ f(x)=\log _{5}(x-3) \] None of the above
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Solution

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Solution Steps

Step 1: Identify the key features of the graph

The graph appears to be a logarithmic function. It passes through the points (4,0) and (8,1). There's a vertical asymptote at x = 3.

Step 2: Determine the base of the logarithm

Since the graph passes through (8,1), we can use this information to help determine the base. We're looking for a base, _b_, such that logb(8-3) = 1, which simplifies to logb(5) = 1. This means b1=5, so b=5.

Step 3: Write the logarithmic function

The general form of a logarithmic function with a horizontal shift is f(x) = logb(x-h), where 'b' is the base and 'h' is the horizontal shift. We determined the base is 5 and the vertical asymptote indicates a horizontal shift of 3 units to the right.

Final Answer

f(x) = log5(x-3)

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