I have a Data that Distribute as Log-Normal.
I do a log transformation in base e. And build a linear model. And after that I backward to original data with $e$$^Y$$e$$^{Var(Y)/2}$. Back transformation With base e
But What if i want to use log in base 10 or 12? How i wiil back transfrom my data?
Thanks.
If you used base $m$ logs instead of base $e$, you just do $m^{\text{whatever}}$ instead of $e^{\text{whatever}}$ for whichever equivalent of the terms or sum of terms you would have exponentiated that's in the original (log-scale) units. If it's in squared units (like a variance) you'd do $m^{\log(m)\text{whatever}}$.
[If it's not obvious that it's really that simple, consider that $\log_m(x)=\log(x)/\log(m)$ and so on the log scale all the log values are linearly scaled versions of the natural log values. So after taking base $m$ logs and doing calculations when it comes to transforming back, consider that you could multiply the estimates and standard errors or standard deviations by $\log(m)$, (giving the corresponding natural log results) and then obviously you can just exponentiate. But $e^{kx}=(e^k)^x$, so $e^{\log(m)x}=m^x$... indicating that indeed all you need to do is take $m$ to the power of whatever things are in the ordinary log-scale units that you got on the $\log_m$ scale rather than the natural log scale. On the other hand things in squared units will have $\log(m)^2$, and only one of those will be taken up by replacing $e$ by $m$.]
This leaved us with a predicted mean of $m^{\hat{y}+\log(m)\sigma^2_\hat{y}/2}$ (however, this is assuming you know $\sigma^2$, otherwise this won't account for the uncertainty in estimating $\sigma^2$ no matter what base you work in).
