Find the Domain y = natural log of x^2+2x

Problem

y=ln(x2+2*x)

Solution

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

x2+2*x>0

2. Factor the quadratic expression to find the critical points where the expression equals zero.

x*(x+2)>0

3. Determine the intervals by finding the roots of the equation x*(x+2)=0 which are x=0 and x=−2

4. Test the intervals (−∞,−2) (−2,0) and (0,∞) to see where the inequality holds true.

• For x=−3 (−3)*(−3+2)=3>0 (True)

• For x=−1 (−1)*(−1+2)=−1<0 (False)

• For x=1 (1)*(1+2)=3>0 (True)

5. State the domain in interval notation based on the intervals that satisfy the inequality.

(−∞,−2)∪(0,∞)

Final Answer

Domain:(−∞,−2)∪(0,∞)

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