I calculated the percent decrease of a value measured at two timepoints:
$\frac{t_1-t_2}{t_1} = d$
but I've now realized that the numbers I was given were log transformed, so it's not really correct to say, for example, that if $d = 0.6$ then the value decreased by 60%, since it's a 60% decrease in log space. I'm trying to figure out if there's a way to transform the $d$ I've already calculated into the correct value without recalculating using the untransformed $t_1$ and $t_2$. It seems like it should be fairly straightforward, but for some reason I'm having a hard time thinking about it. Maybe it's not as simple as I think. Is this an easy thing to do?
Subtract one from the other.
that is $\frac {t_1 - t_2}{t_1} = 1 - \frac {t_2}{t_1}$
If the data has been transformed with a log then $\log t_2 - \log t_1= \log \frac {t_2}{t_1}$
or $1- \exp(\log t_2 - \log t_1) = \frac {t_1 - t_2}{t_1}$
The base of the exponentiation has to match the base of our logs.
And if your logs are natural logs ($ln$) and the percentage changes are fairly small, the mere difference is a very close approximation.
$\ln t_1-\ln t_2 \approx 1- \exp(\ln t_2 - \ln t_1) = \frac {t_1 - t_2}{t_1}$