Solve for x

Problem

(log_2)(x)+(log_2)(x+2)=(log_2)(x+6)

Solution

1. Apply the product rule for logarithms on the left side of the equation, which states (log_b)(M)+(log_b)(N)=(log_b)(M*N)

(log_2)(x*(x+2))=(log_2)(x+6)

2. Use the one-to-one property of logarithms, (log_b)(u)=(log_b)(v)⇒u=v to remove the logs.

x*(x+2)=x+6

3. Distribute the x on the left side to form a quadratic equation.

x2+2*x=x+6

4. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting x and 6 from both sides.

x2+x−6=0

5. Factor the quadratic expression.

(x+3)*(x−2)=0

6. Solve for x by setting each factor to zero.

x=−3,x=2

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_2)(−3) is undefined, x=−3 is extraneous.

x=2

Final Answer

(log_2)(x)+(log_2)(x+2)=(log_2)(x+6)⇒x=2

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