Solve for x 2 log of x = log of 2+ log of 4x-6

Problem

2*(log_)(x)=(log_)(2)+(log_)(4*x−6)

Solution

1. Apply the power property of logarithms to the left side of the equation, which states n*(log_)(a)=(log_)(an)

(log_)(x2)=(log_)(2)+(log_)(4*x−6)

2. Apply the product property of logarithms to the right side of the equation, which states (log_)(a)+(log_)(b)=(log_)(a*b)

(log_)(x2)=(log_)(2*(4*x−6))

3. Distribute the constant inside the logarithm on the right side.

(log_)(x2)=(log_)(8*x−12)

4. Equate the arguments of the logarithms since (log_)(a)=(log_)(b)⇒a=b

x2=8*x−12

5. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 8*x and adding 12 to both sides.

x2−8*x+12=0

6. Factor the quadratic expression by finding two numbers that multiply to 12 and add to −8

(x−6)*(x−2)=0

7. Solve for x by setting each factor equal to zero.

x−6=0⇒x=6

x−2=0⇒x=2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. For x=6 x>0 and 4*(6)−6=18>0 For x=2 x>0 and 4*(2)−6=2>0 Both are valid.

Final Answer

x=6,x=2

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