Expand the Logarithmic Expression natural log of x^2y

Problem

ln(x2*y)

Solution

1. Identify the logarithmic property for products, which states that the log of a product is the sum of the logs of the factors.

ln(a*b)=ln(a)+ln(b)

2. Apply the product rule to the expression by splitting the natural log of x2 and y

ln(x2*y)=ln(x2)+ln(y)

3. Identify the logarithmic property for powers, which states that the log of a base raised to an exponent is the exponent multiplied by the log of the base.

ln(an)=n*ln(a)

4. Apply the power rule to the term ln(x2) by moving the exponent 2 to the front of the logarithm.

ln(x2)+ln(y)=2*ln(x)+ln(y)

Final Answer

ln(x2*y)=2*ln(x)+ln(y)

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