Graph natural log of x^4

Problem

y=ln(x^4)

Solution

1. Identify the domain of the function. Since the argument x^4 must be strictly positive, x^4>0 implies x≠0 The domain is (−∞,0)∪(0,∞)

2. Apply the power rule for logarithms to simplify the expression. Using ln(a^b)=b*ln(a) for even powers requires an absolute value to preserve the domain: y=4*ln(x)

3. Determine the symmetry of the function. Since ƒ*(−x)=ln((−x)^4)=ln(x^4)=ƒ(x) the function is even and symmetric about the yaxis.

4. Find the vertical asymptote. As x approaches 0 from either side, x^4 approaches 0 from the right, so ln(x^4) approaches −∞ There is a vertical asymptote at x=0

5. Identify the xintercepts by setting y=0 Solving ln(x^4)=0 gives x^4=1 which results in x=1 and x=−1

6. Analyze the shape of the graph. For x>0 the graph is a vertically stretched version of the standard natural log curve y=ln(x) For x<0 the graph is a mirror image across the yaxis.

Final Answer

y=ln(x^4)=4*ln(x)

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