Why Returns of Time Series are Stationary?

Almost all econometric models with the time series we use the stationary series for avoiding the sprious regression. When we want to modeling any financial time series, we use the returns of the series. Seemingly we don't take the difference of financial time series for sure the stationarity. In fact we use the differenced financial time series, however some new beginner can't notice the differencing process. In finance, the return series define by the following formula.

                                                            r=log(Y_t/Y_t-1)                                                    (1)

If we apply the properties of the logarithm to this form, we get the following equation.

                                                            r=log(Y_t)-log(Y_t-1)                                           (2)

We can easily see that, equation (2) is the difference of logarithm of Y_t. So, returns of Y_t and ΔYt are same. Thus, after calculate the returns of the any financial time series there is no any requirement to take difference. For this reason every returns series is stationary in mean.