consider the function f(x)=log6(x) a. graph the function.

Answer:

To graph the function (y = \log_{6}(x)), we can use the following steps to find key - points and then plot them.

1. Recall the definition of the logarithmic function (y=\log_{a}(x)) is equivalent to (a^{y}=x) (for (a > 0,a\neq1,x>0)). For (y = \log_{6}(x)), we have (6^{y}=x).
2. Find the x - intercept:
   • Set (y = 0). Then, using the equation (6^{y}=x), when (y = 0), (x=6^{0}=1). So the x - intercept is the point ((1,0)).
3. Find another point:
   • Let (y = 1). Then, using (6^{y}=x), when (y = 1), (x = 6^{1}=6). So we have the point ((6,1)).
   • Let (y=- 1). Then (x=6^{-1}=\frac{1}{6}). So we have the point ((\frac{1}{6}, - 1)).
4. Domain and Asymptote:
   • The domain of (y=\log_{6}(x)) is (x>0). The vertical asymptote is the line (x = 0).
   • Plot the points ((\frac{1}{6},-1)), ((1,0)), ((6,1)) and draw a smooth curve that approaches the vertical asymptote (x = 0) as (x\to0^{+}) and increases as (x) increases.

Explanation:

Step1: Recall logarithmic - exponential equivalence

The function (y = \log_{6}(x)) is equivalent to (6^{y}=x).

Step2: Find x - intercept

Set (y = 0) in (6^{y}=x), so (x = 1).

Step3: Find other points

Set (y = 1) and (y=-1) in (6^{y}=x) to get (x = 6) and (x=\frac{1}{6}) respectively.

Step4: Determine domain and asymptote

Domain is (x>0) and vertical asymptote is (x = 0). Then plot points and draw the curve.