Solve for x 2 log of x = log of 2+ log of 5x-8

Problem

2*(log_)(x)=(log_)(2)+(log_)(5*x−8)

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_)(5*x−8)

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*(5*x−8))

3. Use the one-to-one property of logarithms to remove the logs from both sides, since (log_)(M)=(log_)(N)⇒M=N

x2=2*(5*x−8)

4. Distribute the constant on the right side.

x2=10*x−16

5. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 10*x and adding 16 to both sides.

x2−10*x+16=0

6. Factor the quadratic expression by finding two numbers that multiply to 16 and add to −10

(x−8)*(x−2)=0

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

x−8=0⇒x=8

x−2=0⇒x=2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. For x=8 x>0 and 5*(8)−8>0 For x=2 x>0 and 5*(2)−8>0 Both are valid.

Final Answer

x=2,8

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