Solve for x

Problem

ln(x2+1)−3*ln(x)=ln(2)

Solution

1. Apply the power rule for logarithms to the term 3*ln(x) which states that n*ln(a)=ln(an)

ln(x2+1)−ln(x3)=ln(2)

2. Apply the quotient rule for logarithms to the left side, which states that ln(a)−ln(b)=ln(a/b)

ln((x2+1)/(x3))=ln(2)

3. Exponentiate both sides using the base e to remove the natural logarithms.

(x2+1)/(x3)=2

4. Rearrange the equation into a polynomial form by multiplying both sides by x3

x2+1=2*x3

5. Set the equation to zero to form a cubic equation.

2*x3−x2−1=0

6. Test for rational roots using the Rational Root Theorem. Testing x=1

2*(1)3−(1)2−1=2−1−1=0

7. Factor the polynomial using the known root x=1 by performing synthetic or long division.

(x−1)*(2*x2+x+1)=0

8. Analyze the quadratic factor 2*x2+x+1 using the discriminant D=b2−4*a*c

D=1−4*(2)*(1)=−7

9. Determine valid solutions by noting that the quadratic factor has no real roots and the original logarithmic expression requires x>0 The only real solution is x=1

Final Answer

ln(x2+1)−3*ln(x)=ln(2)⇒x=1

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