Solve for x 2 log of x = log of 2+ log of 6x-10

Problem

2*(log_)(x)=(log_)(2)+(log_)(6*x−10)

Solution

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

(log_)(x^2)=(log_)(2)+(log_)(6*x−10)

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

(log_)(x^2)=(log_)(2*(6*x−10))

3. Simplify the expression inside the logarithm on the right side by distributing the constant.

(log_)(x^2)=(log_)(12*x−20)

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

x^2=12*x−20

5. Rearrange the equation into standard quadratic form by moving all terms to one side.

x^2−12*x+20=0

6. Factor the quadratic equation by finding two numbers that multiply to 20 and add to −12

(x−10)*(x−2)=0

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

x=10

x=2

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. For x=10 10>0 and 6*(10)−10>0 For x=2 2>0 and 6*(2)−10>0 Both are valid.

x=10,2

Final Answer

x=10,2

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