Solve for x

Problem

1/2*ln(x+3)−ln(x)=0

Solution

1. Move the negative term to the other side of the equation to isolate the logarithmic expressions.

1/2*ln(x+3)=ln(x)

2. Apply the power rule of logarithms, a*ln(b)=ln(ba) to the left side of the equation.

ln((x+3)1/2)=ln(x)

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

(x+3)1/2=x

4. Square both sides to eliminate the fractional exponent (the square root).

x+3=x2

5. Rearrange into a quadratic equation by moving all terms to one side.

x2−x−3=0

6. Apply the quadratic formula, x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−1 and c=−3

x=(−(−1)±√(,(−1)2−4*(1)*(−3)))/(2*(1))

7. Simplify the expression under the radical.

x=(1±√(,1+12))/2

x=(1±√(,13))/2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive (x>0 and x+3>0. Since √(,13)≈3.6 the solution (1−√(,13))/2 is negative and must be discarded.

x=(1+√(,13))/2

Final Answer

1/2*ln(x+3)−ln(x)=0⇒x=(1+√(,13))/2

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