Solve for $x$. \[ \log _{2}(x+1)-\log _{2} x=2 \]

Step 1 ：Given the equation \(\log _{2}(x+1)-\log _{2} x=2\)

Step 2 ：Using the properties of logarithms, we can combine the two logarithms on the left side of the equation to get \(\log _{2}\frac{x+1}{x}=2\)

Step 3 ：By definition of logarithms, we can rewrite the equation as \(\frac{x+1}{x}=2^2\)

Step 4 ：Simplifying the right side of the equation gives us \(\frac{x+1}{x}=4\)

Step 5 ：Multiplying both sides of the equation by \(x\) to get rid of the fraction gives us \(x+1=4x\)

Step 6 ：Subtracting \(x\) from both sides to isolate \(x\) on one side of the equation gives us \(1=3x\)

Step 7 ：Finally, dividing both sides by \(3\) to solve for \(x\) gives us \(x=\frac{1}{3}\)

Step 8 ：The solution to the equation is \(\boxed{\frac{1}{3}}\)