Expand the Logarithmic Expression

Problem

ln((8*e2*x5)/((x2+1)√(3,x+2)))

Solution

1. Apply the quotient rule for logarithms, which states that ln(a/b)=ln(a)−ln(b)

ln(8*e2*x5)−ln((x2+1)√(3,x+2))

2. Apply the product rule for logarithms, which states that ln(a*b)=ln(a)+ln(b) to both terms.

ln(8)+ln(e2)+ln(x5)−(ln(x2+1)+ln(√(3,x+2)))

3. Rewrite the radical as a fractional exponent using the property √(n,x)=x(1/n)

ln(8)+ln(e2)+ln(x5)−ln(x2+1)−ln((x+2)(1/3))

4. Apply the power rule for logarithms, which states that ln(ap)=p*ln(a) to move exponents to the front.

ln(8)+2*ln(e)+5*ln(x)−ln(x2+1)−1/3*ln(x+2)

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

ln(8)+2*(1)+5*ln(x)−ln(x2+1)−1/3*ln(x+2)

6. Finalize the expression by combining the constant terms and ensuring all signs are correct.

ln(8)+2+5*ln(x)−ln(x2+1)−1/3*ln(x+2)

Final Answer

ln((8*e2*x5)/((x2+1)√(3,x+2)))=ln(8)+2+5*ln(x)−ln(x2+1)−1/3*ln(x+2)

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