Graph natural log of 1/x

Problem

y=ln(1/x)

Solution

1. Simplify the expression using the properties of logarithms. Since 1/x=x(−1) the expression can be rewritten using the power rule ln(ab)=b*ln(a)

y=ln(x(−1))

y=−ln(x)

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

D:(0,∞)

3. Determine the vertical asymptote. As x approaches 0 from the right, ln(x) approaches −∞ so −ln(x) approaches ∞

Vertical Asymptote:x=0

4. Find the x-intercept by setting y=0

0=−ln(x)

0=ln(x)

e0=x

x=1

5. Analyze the behavior and shape. The graph of y=−ln(x) is a reflection of the standard natural log graph y=ln(x) across the x-axis. As x increases, y decreases.

6. Plot key points to sketch the curve.

Point 1:(1,0)

Point 2:(e,−1)

Point 3:(1/e,1)

Final Answer

y=ln(1/x)=−ln(x)

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