Explanation:

To solve the given differential equation:

- The equation is \(\frac{dy}{dx} = (\ln 5) y\).

- This is a first-order linear ordinary differential equation with separable variables.

- Solve by separating the variables: \(\frac{dy}{y} = \ln 5 \, dx\).

- Integrate both sides: \(\ln |y| = (\ln 5)x + C\).

- Apply the initial condition: \(y(0) = \ln 5\).

- Substitute into the integrated equation: \(\ln (\ln 5) = C\).

- The solution becomes \(y(x) = \ln 5 \cdot e^{(\ln 5)x}\).

- Evaluate \(y(1)\): \(\ln 5 \cdot e^{\ln 5} = (\ln 5) \cdot 5\).

- Thus, \(y(1) = 5 \ln 5\).

Option 4: 5In5 is correct.