Find the Domain log base 2 of cube root of (2+x)/(4x)

Problem

(log_2)(√(3,(2+x)/(4*x)))

Solution

1. Identify the condition for the logarithm to be defined. The argument of a logarithm must be strictly greater than zero.

√(3,(2+x)/(4*x))>0

2. Simplify the inequality. Since the cube root of a number is positive if and only if the number itself is positive, we solve for the radicand.

(2+x)/(4*x)>0

3. Find the critical points where the expression is zero or undefined. These occur when the numerator is zero or the denominator is zero.

2+x=0⇒x=−2

4*x=0⇒x=0

4. Test the intervals created by the critical points x=−2 and x=0 to determine where the rational expression is positive.
   For x<−2 let x=−3 (2−3)/(4*(−3))=(−1)/(−12)>0
   For −2<x<0 let x=−1 (2−1)/(4*(−1))=1/(−4)<0
   For x>0 let x=1 (2+1)/(4*(1))=3/4>0

5. Combine the intervals where the expression is positive to state the domain.

x∈(−∞,−2)∪(0,∞)

Final Answer

Domain=(−∞,−2)∪(0,∞)

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