Question

How do you graph \[y={{\log }_{2}}\left( x-2 \right)\]?

Answer 1

Hint: We know that for a logarithmic function \[{{\log }_{b}}a\] , the value of a is greater than zero and the value of b is greater than zero. For plotting the graph we will substitute some $x$ and find the corresponding value of the function and mark them and join them to get the curve.

Complete step by step answer:
Now from the given question, we were given to draw the graph of \[y={{\log }_{2}}\left( x-2 \right)\].
So, let us compare \[{{\log }_{2}}\left( x-2 \right)\] with \[{{\log }_{b}}a\].
Then it is clear that
\[\begin{align}
  & x-2=a...(1) \\
 & 2=b....(2) \\
\end{align}\]
We know that the value of b should be greater than zero. From equation (2), it is clear that the value of b is equal to 2. We know that 2 is greater than zero.
We know that the value of a should be greater than zero. From equation (1), it is clear that the value of a is equal to \[x-2\]. So, it is clear that
\[\begin{align}
  & \Rightarrow x-2>0 \\
 & \Rightarrow x>2 \\
\end{align}\]
So, it is clear that if the value of x is equal to 2, then the value of y is equal to infinite. We know that if the value of a is equal to 1 in \[{{\log }_{b}}a\], then the value of \[{{\log }_{b}}a\] is equal to 0.
So, if
\[\begin{align}
  & \Rightarrow x-2=1 \\
 & \Rightarrow x=3 \\
\end{align}\]
So, it is clear that if the value of x is equal to 3, then the value of y is equal to 0.
Now let us substiute the value of x is equal to 4.
\[\Rightarrow y={{\log }_{2}}\left( 4-2 \right)={{\log }_{2}}2=1\].
So, it is clear that if the value of x is equal to 4, then the value of y is equal to 1.
So, the points \[\left( 2,\infty \right),\left( 3,0 \right),\left( 4,1 \right)\] lie on the curve \[y={{\log }_{2}}\left( x-2 \right)\]. Now let us join these three points to have the graph of the equationn \[y={{\log }_{2}}\left( x-2 \right)\].

The above graph shows us the graph of \[y={{\log }_{2}}\left( x-2 \right)\].

Note: We should be very careful while plotting the points in the graph. Also, we should be very careful while finding the points to be plotted in the graph. We should be very careful while drawing the graph. Also, we should be very well known about drawing the graphs of logarithms. Similarly we can plot the graph of any logarithmic functions.