graph the logarithmic function ( y=log _{6}(x + 5)-4 ). first, graph the vertical asymptote followed by two…

Explanation:

Step1: Find the vertical asymptote

For a logarithmic function (y = \log_{b}(x - h)+k), the vertical asymptote is (x = h). In the function (y=\log_{6}(x + 5)-4), we have (h=- 5). So the vertical asymptote is (x=-5).

Step2: Find two points

• Point 1: Let (x=-4) Substitute (x = - 4) into (y=\log_{6}(x + 5)-4). (y=\log_{6}(-4 + 5)-4=\log_{6}(1)-4) Since (\log_{a}(1)=0) for (a>0,a\neq1), then (y = 0-4=-4). So the point is ((-4,-4))
• Point 2: Let (x = 1) Substitute (x = 1) into (y=\log_{6}(x + 5)-4). (y=\log_{6}(1 + 5)-4=\log_{6}(6)-4) Since (\log_{a}(a)=1) for (a>0,a\neq1), then (y=1 - 4=-3). So the point is ((1,-3))

Answer:

Vertical asymptote (x = - 5), points ((-4,-4)) and ((1,-3))