Graph natural log of x^3

Problem

y=ln(x3)

Solution

1. Identify the domain of the function. Since the argument of a natural logarithm must be positive, we require x3>0 which implies x>0

2. Apply logarithm properties to simplify the expression. Using the power rule ln(ab)=b*ln(a) the function becomes y=3*ln(x)

3. Determine the vertical asymptote by finding where the argument is zero. As x→0 y→−∞ so there is a vertical asymptote at x=0

4. Find the x-intercept by setting y=0 Solving 3*ln(x)=0 gives ln(x)=0 which means x=e0=1 The intercept is at (1,0)

5. Analyze the behavior and shape. Since the derivative d(y)/d(x)=3/x is always positive for x>0 the graph is strictly increasing. The second derivative d2(y)/(d(x)2)=−3/(x2) is always negative, meaning the graph is concave down.

6. Plot key points to sketch the curve. For x=e y=3*ln(e)=3 For x=1/e y=3*ln(e(−1))=−3

Final Answer

y=ln(x3)=3*ln(x)* for *x>0

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