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  • $\begingroup$ It is not only the effect of "log-transform", but also the differential operator! To me, the second form is just a math equivalence (perhaps easier to remember) to the first form, whose intuition is as yourself stated. $\endgroup$ Commented Feb 11 at 2:24
  • $\begingroup$ By the way, the expression you listed had an obvious typo -- there should not be a negative sign in any of expressions. $\endgroup$ Commented Feb 11 at 2:27