Find the Domain f(x) = log base 7 of x^2-9

Problem

ƒ(x)=(log_7)(x2−9)

Solution

1. Identify the condition for the domain of a logarithmic function. The argument of the logarithm must be strictly greater than zero.

x2−9>0

2. Factor the quadratic expression using the difference of squares formula.

(x−3)*(x+3)>0

3. Determine the critical points by setting each factor to zero.

x=3

x=−3

4. Test the intervals created by the critical points: (−∞,−3) (−3,3) and (3,∞)
   For x<−3 the product is positive.
   For −3<x<3 the product is negative.
   For x>3 the product is positive.

5. Select the intervals where the expression is positive to satisfy the inequality.

(−∞,−3)∪(3,∞)

Final Answer

Domain:(−∞,−3)∪(3,∞)

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