Evaluate natural log of square root of e^3

Problem

ln(√(,e^3))

Solution

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

ln((e^3)^(1/2))

2. Apply the power of a power rule for exponents, which states (a^m)^n=a^(m⋅n)

ln(e^(3/2))

3. Use the logarithm power rule ln(x^k)=k*ln(x) to move the exponent in front of the natural log.

3/2*ln(e)

4. Evaluate the natural log of e knowing that ln(e)=1

3/2⋅1

5. Simplify the resulting value.

3/2

Final Answer

ln(√(,e^3))=3/2

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