Expand the Logarithmic Expression natural log of e^(2/5)x^7y^(3/2)

Problem

ln(e(2/5)*x7*y(3/2))

Solution

1. Apply the product rule for logarithms, which states that ln(a*b*c)=ln(a)+ln(b)+ln(c)

ln(e(2/5)*x7*y(3/2))=ln(e(2/5))+ln(x7)+ln(y(3/2))

2. Apply the power rule for logarithms, which states that ln(an)=n*ln(a) to each term.

ln(e(2/5))+ln(x7)+ln(y(3/2))=2/5*ln(e)+7*ln(x)+3/2*ln(y)

3. Simplify the natural log of e using the identity ln(e)=1

2/5*(1)+7*ln(x)+3/2*ln(y)

4. Combine the terms into the final expanded form.

2/5+7*ln(x)+3/2*ln(y)

Final Answer

ln(e(2/5)*x7*y(3/2))=2/5+7*ln(x)+3/2*ln(y)

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