Expand the Logarithmic Expression natural log of cube root of 6r^5s

Problem

ln(√(3,6*r^5*s()))

Solution

1. Rewrite the radical expression using a fractional exponent.

ln(6*r^5*s)

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

1/3*ln(6*r^5*s)

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

1/3*(ln(6)+ln(r^5)+ln(s))

4. Apply the power rule again to the term ln(r^5)

1/3*(ln(6)+5*ln(r)+ln(s))

5. Distribute the fraction to each term in the expression.

1/3*ln(6)+5/3*ln(r)+1/3*ln(s)

Final Answer

ln(√(3,6*r^5*s()))=1/3*ln(6)+5/3*ln(r)+1/3*ln(s)

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