Find the Domain

Problem

(log_)(√(,7*x+8)/((3*x+2)6√(4,2*x−5)))

Solution

1. Identify the constraints for the logarithm function. The argument of a logarithm must be strictly greater than zero:

√(,7*x+8)/((3*x+2)6√(4,2*x−5))>0

2. Identify the constraints for the even-indexed roots. The radicand of a square root and a fourth root must be non-negative:

7*x+8≥0

2*x−5≥0

3. Identify the constraints for the denominator. The denominator cannot be equal to zero:

(3*x+2)6≠0

√(4,2*x−5)≠0

4. Solve the inequality for the first root. Since the root is in the numerator and the entire expression must be positive, the radicand must be strictly positive to avoid a zero numerator:

7*x+8>0

7*x>−8

x>−8/7

5. Solve the inequality for the second root. Since the fourth root is in the denominator, the radicand must be strictly positive to avoid division by zero:

2*x−5>0

2*x>5

x>5/2

6. Check the remaining denominator term. The term (3*x+2)6 is zero when x=−2/3 However, we must satisfy all conditions simultaneously.

7. Determine the intersection of all conditions. We compare x>−8/7 and x>5/2 Since 5/2=2.5 and −8/7≈−1.14 the condition x>5/2 is more restrictive.

8. Verify the intersection. If x>5/2 then x is never −2/3 so the denominator term (3*x+2)6 is always positive and non-zero.

Final Answer

Domain=(5/2,∞)

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