relchange <- function(x, first = FALSE) {
x[!is.na(x)] <- c(1, exp(diff(log(x[!is.na(x)]))))
return(x)
}
relchange(dt$sale)
[1] NA NA NA NA 1.0000000 0.6930112 1.3048314
[8] 1.4211981 5.3062771 1.0084781 1.1261770 1.1349657