Evaluate natural log of square root of e^6

Problem

ln(√(,e6))

Solution

1. Rewrite the square root as a fractional exponent using the property √(,x)=x(1/2)

ln((e6)(1/2))

2. Simplify the exponent by applying the power of a power rule (am)n=a(m⋅n)

6⋅1/2=3

ln(e3)

3. Apply the property of logarithms ln(ex)=x because the natural logarithm and the exponential function are inverses.

ln(e3)=3

Final Answer

ln(√(,e6))=3

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