## Precalculus (6th Edition) Blitzer Draw the graph of $f\left( x \right)=\ln x$ as follows: Construct the table of coordinates for $f\left( x \right)={{\ln } }x$ 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, using the $y\text{-axis}$ or $x=0$ as the vertical asymptote. Now, the curve of $g\left( x \right)=-\ln \left( 2x \right)$ in the graph above is obtained by horizontally shrinking the curve of $f\left( x \right)=\ln x$ by a factor of 2 and then taking the reflection about the $x\text{-axis}$. Divide the each x-coordinate by 2 units in the rectangular coordinate system. Multiply the y-coordinate by $-1$ and the curve $g\left( x \right)=-\ln \left( 2x \right)$ is obtained as a reflection about the x-axis in the rectangular plane. Observing the above graph: The vertical asymptote is $x=0$, domain is $\left( -\infty,0 \right)$, and range is $\left( -\infty,\infty \right)$ of the function $f\left( x \right)=\ln x$ The vertical asymptote is $x=0$, domain is $\left( -\infty,0 \right)$, and range is $\left( -\infty,\infty \right)$ of the function $g\left( x \right)=-\ln \left( 2x \right)$.