Evaluate natural log of square root of ab

Problem

ln(√(,a*b))

Solution

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

ln((a*b)(1/2))

2. Apply the power rule for logarithms, which states ln(xk)=k*ln(x)

1/2*ln(a*b)

3. Apply the product rule for logarithms, which states ln(x*y)=ln(x)+ln(y)

1/2*(ln(a)+ln(b))

4. Distribute the constant factor to each term in the expression.

1/2*ln(a)+1/2*ln(b)

Final Answer

ln(√(,a*b))=(ln(a)+ln(b))/2

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