Question

graph each of the following log functions. to graph them, first select the func its asymptote, and finally selecting two points on the graph.
$f(x)=\log _{5}(x - 1)$
$f(x)=\log _{5}(x)-2$

Explanation:

Step1: Find the asymptote for \(f(x)=\log_{5}(x - 1)\)

For a logarithmic function \(y=\log_{a}(x - h)+k\), the vertical asymptote is \(x = h\). Here \(h = 1\), so the vertical asymptote is \(x=1\).

Step2: Find two points for \(f(x)=\log_{5}(x - 1)\)

• When \(x=2\), \(f(2)=\log_{5}(2 - 1)=\log_{5}(1)=0\). So the point is \((2,0)\).
• When \(x=6\), \(f(6)=\log_{5}(6 - 1)=\log_{5}(5)=1\). So the point is \((6,1)\).

Step3: Find the asymptote for \(f(x)=\log_{5}(x)-2\)

For the function \(y=\log_{a}(x)+k\), the vertical asymptote is \(x = 0\) (since \(h = 0\) in the form \(y=\log_{a}(x - h)+k\)).

Step4: Find two points for \(f(x)=\log_{5}(x)-2\)

• When \(x = 1\), \(f(1)=\log_{5}(1)-2=0 - 2=-2\). So the point is \((1,-2)\).
• When \(x = 5\), \(f(5)=\log_{5}(5)-2=1 - 2=-1\). So the point is \((5,-1)\).

Answer:

For \(f(x)=\log_{5}(x - 1)\):

• Asymptote: \(x = 1\)
• Points: \((2,0)\) and \((6,1)\)

For \(f(x)=\log_{5}(x)-2\):

• Asymptote: \(x = 0\)
• Points: \((1,-2)\) and \((5,-1)\)