Evaluate natural log of square root of e^3

Problem

ln(√(,e3))

Solution

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

ln((e3)(1/2))

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

ln(e(3/2))

3. Use the logarithm power rule ln(xk)=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(√(,e3))=3/2

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