Graph y = natural log of x

Problem

y=ln(x)

Solution

1. Identify the domain of the function. Since the natural logarithm is only defined for positive values, the domain is x>0 This implies a vertical asymptote at x=0

2. Determine the x-intercept by setting y=0 Solving 0=ln(x) gives e0=x so the graph passes through (1,0)

3. Calculate key points to determine the shape. When x=e≈2.718 y=ln(e)=1 When x=1/e≈0.368 y=ln(1/e)=−1

4. Analyze the behavior as x approaches zero and infinity. As x→0 y→−∞ As x→∞ y→∞ at a decreasing rate.

5. Sketch the curve starting from the vertical asymptote x=0 passing through (1,0) and (e,1) and continuing to rise slowly to the right.

Final Answer

The graph of y=ln(x) is a curve in the first and fourth quadrants with a vertical asymptote at x=0 an x-intercept at (1,0) and passing through the point (e,1)

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