Solve for $y$ in terms of $x$. \[ \ln y+2 \ln x=4+\ln 10 \]

Step 1 ：Given the equation \(\ln y+2 \ln x=4+\ln 10\)

Step 2 ：Combine the logarithms on the left side of the equation using the property of logarithms that states \(\ln a + \ln b = \ln (ab)\) to get \(\ln(xy^2) = 4 + \ln 10\)

Step 3 ：Subtract \(\ln 10\) from both sides to get \(\ln(xy^2) = 4\)

Step 4 ：Use the property of logarithms that states \(\ln a = b\) is equivalent to \(a = e^b\) to get \(xy^2 = e^4\)

Step 5 ：Divide both sides by \(x^2\) to isolate \(y\)

Step 6 ：Final Answer: The solution for \(y\) in terms of \(x\) is \(y = \frac{10e^4}{x^2}\). So, \(y = \boxed{\frac{10e^4}{x^2}}\)