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So an additive relationship on the log scale translates to a multiplicative relationship on the natural scale. But what does a multiplicative relationship on the log scale translate to on the natural scale.

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The premise of this question is wrong. It is a multiplicative relationship on the natural scale that justifies the use of log-transform to estimate additive effects.

For instance:

$z = a b ^ c$

implies

$\log(z) = \log(a) + c \log(b)$

by arithmetic rules.

Additive effects do not have a known relational scale when log transformed, but approximations may be applied using Taylor expansions to obtain other additive effects.

For instance $\log(1+a) \approx a $ when $a$ very close to 0.

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If you have $\log(y) = ab$ then $y = e^{ab}$.

If either of the terms $a$ or $b$ is on the log scale, it converts to a power.

i.e. if $\log(y) = a \log (x)$ then $y = x^a$

Note that we've ignored what happens to the error term; in practice you must consider what's going on with that.

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