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Given a logarithmic function with the form $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x\right),$ graph the function.
Graph $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{5}\left(x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
Before graphing, identify the behavior and key points for the graph.
The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$
Graph $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{\frac{1}{5}}(x).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$
As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ without loss of shape.
When a constant $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is added to the input of the parent function $\text{\hspace{0.17em}}f(x)=lo{g}_{b}(x),$ the result is a horizontal shift $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units in the opposite direction of the sign on $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ To visualize horizontal shifts, we can observe the general graph of the parent function $f(x)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and for $\text{\hspace{0.17em}}c>0\text{\hspace{0.17em}}$ alongside the shift left, $\text{\hspace{0.17em}}g(x)={\mathrm{log}}_{b}\left(x+c\right),$ and the shift right, $\text{\hspace{0.17em}}h(x)={\mathrm{log}}_{b}\left(x-c\right).$ See [link] .
For any constant $\text{\hspace{0.17em}}c,$ the function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x+c\right)$
Given a logarithmic function with the form $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}\left(x+c\right),$ graph the translation.
Sketch the horizontal shift $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x-2)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
Since the function is $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x-2),$ we notice $\text{\hspace{0.17em}}x+\left(-2\right)=x\u20132.$
Thus $\text{\hspace{0.17em}}c=-2,$ so $\text{\hspace{0.17em}}c<0.\text{\hspace{0.17em}}$ This means we will shift the function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{3}(x)\text{\hspace{0.17em}}$ right 2 units.
The vertical asymptote is $\text{\hspace{0.17em}}x=-(-2)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=2.$
Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{3},\mathrm{-1}\right),$ $\left(1,0\right),$ and $\text{\hspace{0.17em}}\left(3,1\right).$
The new coordinates are found by adding 2 to the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinates.
Label the points $\text{\hspace{0.17em}}\left(\frac{7}{3},\mathrm{-1}\right),$ $\left(3,0\right),$ and $\text{\hspace{0.17em}}\left(5,1\right).$
The domain is $\text{\hspace{0.17em}}\left(2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=2.$
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