Graph natural log of x^2

Problem

y=ln(x2)

Solution

1. Identify the domain of the function. Since the argument of a logarithm must be positive, we require x2>0 which means x≠0 The domain is (−∞,0)∪(0,∞)

2. Apply logarithm properties to simplify the expression. Using the power rule ln(ab)=b*ln(a) for even powers, we rewrite the function as y=2*ln(x)

3. Determine symmetry by checking if the function is even or odd. Since ƒ*(−x)=ln((−x)2)=ln(x2)=ƒ(x) the function is even and symmetric about the yaxis.

4. Find the vertical asymptote by looking at where the argument of the logarithm approaches zero. As x→0 y→−∞ so there is a vertical asymptote at x=0

5. Find the x-intercepts by setting y=0 Solving ln(x2)=0 gives x2=e0=1 resulting in intercepts at (1,0) and (−1,0)

6. Analyze the end behavior as x increases or decreases. As x→∞ or x→−∞ y→∞ The graph rises slowly on both sides of the yaxis.

Final Answer

y=ln(x2)

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