Simplify/Condense natural log of square root of ex

Problem

ln(√(,e*x))

Solution

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

ln((e*x)(1/2))

2. Apply the power rule for logarithms, ln(ab)=b*ln(a) to move the exponent to the front.

1/2*ln(e*x)

3. Apply the product rule for logarithms, ln(a*b)=ln(a)+ln(b) to separate the terms inside the natural log.

1/2*(ln(e)+ln(x))

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

1/2*(1+ln(x))

5. Distribute the constant to reach the final simplified form.

1/2+1/2*ln(x)

Final Answer

ln(√(,e*x))=(1+ln(x))/2

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