Begin by graphing f(x)=log_{2}x Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each functions domain and range. g(x)= -2log_{2}x

Step 1 For the following Logarithmic function defined by 1) ${\displaystyle f\left(x\right)={\log }_{2}x}$ Graphing Logarithmic function given in equation (1) requires setting up table of coordinates, so that

$\begin{array}{|ccc|}\hline x& f(x)=y={\log }_{2}x& (x,y)\\ 1& f(1)=y={\log }_2(1)\Rightarrow 2^y=1=2^0\Rightarrow y=0& (1,0)\\ 2& f(2)=y={\log }_2(2)\Rightarrow 2^y=2=2^1\Rightarrow y=1& (2,1)\\ 4& f(4)=y={\log }_2(4)\Rightarrow 2^y=4=2^2\Rightarrow y=2& (4,2)\\ 8& f(8)=y={\log }_2(8)\Rightarrow 2^y=8=2^3\Rightarrow y=3& (8,3)\\ 16& f(16)=y={\log }_2(16)\Rightarrow 2^y=16=2^4\Rightarrow y=4& (16,4)\\ 32& f(32)=y={\log }_2(32)\Rightarrow 2^y=32=2^5\Rightarrow y=5& (32,5)\\ 64& f(64)=y={\log }_2(64)\Rightarrow 2^y=64=2^6\Rightarrow y=6& (64,6)\\ \hline\end{array}$

Step 2 We plot the following points between (x, y) determined from the table of coordinates and connect them with the continuous curve which represent the Logarithmic function ${\displaystyle f\left(x\right)={\log }_{2}x}$ as shown in Figure (1). Figure (1) represent the graph of Logarithmic function $f(x)={\log }_{2}x$ 

Step 3 Note that: The y-axis which represented by the equation ${\displaystyle x=0}$ is the vertical asymptote, so that the curve approches, but never touches the positive portion of the y-axis as shown in figure (1). The domain of $f(x)={\log }_2(x)\text{is all positive real numbers}x\in (0,\mathrm{\infty })\text{andthe range is all real numbers}y\in (-\mathrm{\infty },\mathrm{\infty })$.

Step 4 To graph the Logarithmic function