Solve for \( x \) \[ \log _{4}(x-4)-\log _{4} 2=\log _{4} x \]

To solve the equation \[ \log _{4}(x-4)-\log _{4} 2=\log _{4} x, \] we can use the properties of logarithms. Recall that the difference of logarithms can be expressed as a logarithm of a quotient: \[ \log_{4}(x-4) - \log_{4}(2) = \log_{4}\left(\frac{x-4}{2}\right). \] Thus, we can rewrite the equation as: \[ \log_{4}\left(\frac{x-4}{2}\right) = \log_{4} x. \] Since the logarithms are equal, we can set the arguments equal to each other: \[ \frac{x-4}{2} = x. \] Next, we can clear the fraction by multiplying both sides by 2: \[ x - 4 = 2x. \] Now, we isolate \(x\) by subtracting \(x\) from both sides: \[ -4 = 2x - x, \] which simplifies to \[ -4 = x. \] Now we have a potential solution, but we should check whether it is valid in the original logarithmic expressions. To ensure the logarithms are defined, we must have: 1. \(x - 4 > 0\) implies \(x > 4\). 2. \(x > 0\). Since our solution \( x = -4 \) does not satisfy \( x > 4 \), it is not a valid solution. Given this analysis, there are no solutions for the equation.