Solve for x log of 4x = log of 5+ log of x-3

Problem

(log_)(4*x)=(log_)(5)+(log_)(x−3)

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_)(4*x)=(log_)(5*(x−3))

2. Distribute the constant 5 inside the parentheses on the right side.

(log_)(4*x)=(log_)(5*x−15)

3. Use the property of equality for logarithms, which states that if (log_)(u)=(log_)(v) then u=v

4*x=5*x−15

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

−x=−15

5. Solve for x by multiplying both sides by −1

x=15

6. Verify the solution by checking if the arguments of the original logarithms are positive. Since 4*(15)>0 and 15−3>0 the solution is valid.

Final Answer

(log_)(4*x)=(log_)(5)+(log_)(x−3)⇒x=15

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