Solve for x log of 3x = log of 5+ log of x-2

Problem

(log_)(3*x)=(log_)(5)+(log_)(x−2)

Solution

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

(log_)(3*x)=(log_)(5*(x−2))

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

(log_)(3*x)=(log_)(5*x−10)

3. Use the one-to-one property of logarithms, which states that if (log_)(u)=(log_)(v) then u=v

3*x=5*x−10

4. Isolate the variable by subtracting 5*x from both sides of the equation.

−2*x=−10

5. Solve for x by dividing both sides by −2

x=5

6. Verify the solution by checking if the arguments of the original logarithms are positive. Since 3*(5)>0 and 5−2>0 the solution is valid.

Final Answer

(log_)(3*x)=(log_)(5)+(log_)(x−2)⇒x=5

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