Which of the following is equivalent to $2 \ln (10 x)$ for $x>0$ ? Choose the correct answer below. $\ln 100+\ln x$ $\ln (20 x)$ $\ln 20+\ln x$ $\ln \left(100 x^{2}\right)$

Step 1 ：Given the expression \(2 \ln (10 x)\) for \(x>0\).

Step 2 ：Using the property of logarithms that states \(n \ln a = \ln a^n\), we can rewrite the given expression as \(\ln (10x)^2\).

Step 3 ：This simplifies to \(\ln (100x^2)\).

Step 4 ：So, the equivalent expression to \(2 \ln (10 x)\) for \(x>0\) is \(\ln \left(100 x^{2}\right)\).

Step 5 ：\(\boxed{\ln \left(100 x^{2}\right)}\) is the final answer.