Graph natural log of natural log of x^2

Problem

ln(ln(x2))

Solution

1. Identify the domain of the function ƒ(x)=ln(ln(x2)) For the outer natural log to be defined, the inner argument must be positive: ln(x2)>0

2. Solve the inequality ln(x2)>0 by exponentiating both sides, which gives x2>e0 or x2>1 This implies the domain is (−∞,−1)∪(1,∞)

3. Determine the vertical asymptotes by finding where the inner argument ln(x2) approaches zero. This occurs at x=1 and x=−1 As x→1 or x→−1 ƒ(x)→−∞

4. Analyze the symmetry of the function. Since ƒ(x)=ƒ*(−x) the graph is symmetric about the yaxis (an even function).

5. Find the x-intercepts by setting ƒ(x)=0 This occurs when ln(x2)=1 which means x2=e or x=±√(,e)≈±1.649

6. Observe the end behavior. As x→∞ or x→−∞ the function ƒ(x) grows without bound, though very slowly due to the nested logarithms.

Final Answer

ƒ(x)=ln(ln(x2))

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