Answer
See the graph below:
Work Step by Step
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)$.
