Begin with the graph of $f(x)=\log _{2} x$ and use transformations to sketch the graph of the function. Find the domain and range of the function and the vertical asymptote of the graph. \[ y=\log _{2}(x-2) \] Use the graphing tool to graph the equation.
Solution
Step 1 ：The function \(y=\log _{2}(x-2)\) is a transformation of the function \(f(x)=\log _{2} x\). Specifically, it is a horizontal shift of the graph of \(f(x)\) to the right by 2 units.
Step 2 ：The domain of the function \(y=\log _{2}(x-2)\) is all real numbers greater than 2, because the logarithm function is only defined for positive numbers and \(x-2\) must be positive.
Step 3 ：The range of the function is all real numbers, because the logarithm function can take any real number as its output.
Step 4 ：The vertical asymptote of the graph is the line \(x=2\), because as \(x\) approaches 2 from the right, \(y=\log _{2}(x-2)\) approaches negative infinity.
Step 5 ：\(\boxed{\text{The domain of the function } y=\log _{2}(x-2) \text{ is } (2, \infty), \text{ the range is } (-\infty, \infty), \text{ and the vertical asymptote is the line } x=2}\)
