Graph natural log of x^8

Problem

y=ln(x8)

Solution

1. Identify the domain of the function. Since the argument of a natural logarithm must be positive, we require x8>0 This is true for all real numbers except x=0 Thus, the domain is (−∞,0)∪(0,∞)

2. Simplify the expression using the power rule for logarithms, ln(ab)=b*ln(a) However, because the exponent 8 is even, we must use an absolute value to preserve the domain: y=8*ln(x)

3. Determine the symmetry of the function. Since ƒ*(−x)=ln((−x)8)=ln(x8)=ƒ(x) the function is even. This means the graph is symmetric with respect to the yaxis.

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

5. Identify the x-intercepts by setting y=0

ln(x8)=0

x8=e0

x8=1

x=1,x=−1

6. Analyze the behavior as x increases. As |x| grows, 8*ln(x) grows slowly toward infinity. The graph will look like a standard natural log curve scaled vertically by a factor of 8, mirrored across the yaxis.

Final Answer

y=8*ln(x)

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