Graph natural log of xy

Problem

ln(x*y)

Solution

1. Identify the domain of the function. For the natural logarithm to be defined, the argument must be strictly positive, so x*y>0 This occurs when both x and y are positive (Quadrant I) or both x and y are negative (Quadrant III).

2. Apply the logarithm product rule to rewrite the expression as ƒ(x,y)=ln(x)+ln(y)

3. Determine the nature of the graph. Since this is a function of two variables, the graph is a surface in three-dimensional space, z=ln(x)+ln(y)

4. Analyze the traces to visualize the surface. Setting z=k (where k is a constant) gives the level curves ln(x*y)=k which simplifies to x*y=ek These level curves are hyperbolas in the x*yplane.

5. Observe the behavior near the axes. As x→0 or y→0 the value of z→−∞ indicating vertical asymptotic behavior along the x and y axes.

Final Answer

z=ln(x)+ln(y)

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