Find the largest value of $x$ that satisfies: \[ \begin{array}{l} \log _{3}\left(x^{2}\right)-\log _{3}(x+1)=2 \\ x= \end{array} \] You may enter the exact value or round to 4 decin

Step 1 ：The given equation is in the form of a logarithmic equation. We can use the properties of logarithms to simplify the equation and solve for x. The property we will use is that the difference of two logarithms is equal to the logarithm of the quotient of their arguments. So, we can rewrite the equation as: \(\log _{3}\left(\frac{x^{2}}{x+1}\right)=2\)

Step 2 ：Then, we can convert the logarithmic equation to an exponential equation. The base is 3, the exponent is 2, and the result is the argument of the logarithm. So, we get: \(\frac{x^{2}}{x+1}=3^{2}\)

Step 3 ：Solving this equation will give us the value of x. The solutions for x are \(9/2 - 3\sqrt{13}/2\) and \(9/2 + 3\sqrt{13}/2\). These are the two values of x that satisfy the given equation.

Step 4 ：However, the question asks for the largest value of x. Therefore, the largest value of x is \(9/2 + 3\sqrt{13}/2\).

Step 5 ：Final Answer: The largest value of x that satisfies the given equation is \(\boxed{9/2 + 3\sqrt{13}/2}\).