Evaluate natural log of square root of e^6

Problem

ln(√(,e^6))

Solution

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

ln((e^6)^(1/2))

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

6⋅1/2=3

ln(e^3)

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

ln(e^3)=3

Final Answer

ln(√(,e^6))=3

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