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I know that the geometric mean is used in order to make percentage returns across time comparable. Similarly, I know that log prices make percentage returns comparable for example when prices are charted.

What is the mathematical basis for this relationship? Put differently, in what way do log returns "correspond" to geometric means and in what way to simple returns "correspond" to arithmetic means?

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If you wanted to see the following (price $S_t$, log return $r$, simple return $R$) then $$ r = \log(S_{t+1}) - \log(S_t) = \log(S_{t+1}/S_{t}), $$ and $$ R = S_{t+1}/S_{t}-1, $$ thus $$ R = \exp(r)-1 $$ and $$r = \log(1+R).$$ Was this the question?

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