Graph natural log of square root of x

Problem

y=ln(√(,x))

Solution

1. Simplify the expression using the properties of logarithms. Since √(,x)=x(1/2) the exponent can be moved to the front of the natural log.

y=ln(x(1/2))

y=1/2*ln(x)

2. Identify the domain of the function. Because the argument of a logarithm must be positive, the domain is x>0

3. Determine the vertical asymptote. As x approaches 0 from the right, ln(x) approaches −∞ so there is a vertical asymptote at x=0

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

0=1/2*ln(x)

0=ln(x)

e0=x

x=1

5. Calculate key points to determine the shape of the curve.

If *x=e,y=1/2*ln(e)=1/2

If *x=e2,y=1/2*ln(e2)=1

6. Sketch the graph. The function is a logarithmic curve that passes through (1,0) and (e,0.5) increasing slowly as x increases, and approaching the y-axis as a vertical asymptote.

Final Answer

y=ln(√(,x))=1/2*ln(x)

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