Answer
See the graph below:
Work Step by Step
We use the rule of logarithm ${{\log }_{b}}{{b}^{z}}=z $.
Construct the table of coordinates for $ f\left( x \right)={{\log }_{2}}x $ and choose the appropriate values for $ x $ and calculate the corresponding $ y\text{-values}$.
Now, plot every point provided in the above table and join them using a smooth curve, and the $ y\text{-axis}$ or $ x=0$ is the vertical asymptote.
Now, using transformation, the curve of $ g\left( x \right)={{\log }_{2}}\left( x-2 \right)$ in the graph is similar to the graph of $ f\left( x \right)={{\log }_{2}}x $ translated horizontally to the right by 2 units.
So, shift the curve of $ f\left( x \right)={{\log }_{2}}x $ to the right by 2 units.
Also, in the graph, the vertical asymptote also shifts 2 units to the right to become $ x=2$.
Now, observe in the graph of $ g\left( x \right)={{\log }_{2}}\left( x-2 \right)$, x-axis includes all real values greater than 2, and the range includes all real numbers.
So, the domain of the stated function $ f\left( x \right)={{\log }_{2}}x $ is $\left( 2,\infty \right)$, and the range is $\left( -\infty,\infty \right)$.
Thus, after observing the graph, the $ x\text{-intercept}$ of the function $ g\left( x \right)={{\log }_{2}}\left( x-2 \right)$ is $\left( 3,0 \right)$, the vertical asymptote is $ x=2$, the domain of the provided function is $\left( 2,\infty \right)$, and the range is $\left( -\infty,\infty \right)$.
