6.4 Graphs of logarithmic functions  (Page 5/8)

Since the function is $f(x)=2{\log }_4(x),$ we will notice $a=2.$

This means we will stretch the function $f(x)={\log }_4(x)$ by a factor of 2.

The vertical asymptote is $x=0.$

Consider the three key points from the parent function, $\left(\frac{1}{4},\mathrm{-1}\right),$ $(1,0),$ and $\left(4,1\right).$

The new coordinates are found by multiplying the $y$ coordinates by 2.

Label the points $\left(\frac{1}{4},\mathrm{-2}\right),$ $\left(1,0\right),$ and $\left(4,\text{2}\right).$

The domain is $\left(\mathrm{0,}\infty \right),$ the range is $(-\infty ,\infty ),$ and the vertical asymptote is $x=0.$ See [link] .

The domain is $\left(0,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=0.$

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to $x=\mathrm{-2.}$ The x -intercept will be $(\mathrm{-1,}0).$ The domain will be $\left(\mathrm{-2},\infty \right).$ Two points will help give the shape of the graph: $(\mathrm{-1},0)$ and $(8,5).$ We chose $x=8$ as the x -coordinate of one point to graph because when $x=\mathrm{8,}$ $x+2=\mathrm{10,}$ the base of the common logarithm.

The domain is $\left(-2,\infty \right),$ the range is $\left(-\infty ,\infty \right),$ and the vertical asymptote is $x=-2.$