Solve for x log of x+4- log of x = log of x+2

Problem

(log_)(x+4)−(log_)(x)=(log_)(x+2)

Solution

1. Apply the quotient rule for logarithms to the left side of the equation, which states that (log_)(a)−(log_)(b)=(log_)(a/b)

(log_)((x+4)/x)=(log_)(x+2)

2. Remove the logarithms by using the property that if (log_)(A)=(log_)(B) then A=B

(x+4)/x=x+2

3. Multiply both sides by x to eliminate the fraction.

x+4=x*(x+2)

4. Distribute the x on the right side of the equation.

x+4=x2+2*x

5. Rearrange into a quadratic equation by subtracting x and 4 from both sides to set the equation to zero.

0=x2+x−4

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

x=(−1±√(,1−4*(1)*(−4)))/(2*(1))

7. Simplify the discriminant and the expression.

x=(−1±√(,1+16))/2

x=(−1±√(,17))/2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since x must be greater than 0 we must reject the negative root.

x=(−1+√(,17))/2

Final Answer

(log_)(x+4)−(log_)(x)=(log_)(x+2)⇒x=(−1+√(,17))/2

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