Formula used:\(\int \frac{dy}{y} = ln \ y + C\)log m + log n = log mnlog mn = n log m Calculation:Given that, \(\frac{dy}{dx} = (\ln 5)y \)? \(\frac{dy}{y} = (\ln 5)dx\)Doing integration on both sides? \(\int\frac{dy}{y} =\int (\ln 5)dx\)Since, \(\int \frac{dy}{y} = ln \ y + C\) ? ln y = x ln 5 + ln CUsing the logrethamic property (2) & (3)? ln y = ln 5x + ln C? ln y = ln C5x? y = C 5x ------(1)According to the question, y(0) = ln 5ln 5 = C 50 ? C = ln 5Put this value in equation (1)? y = ln 5 5xPut x = 1? y(1) = 5 ln 5