# Timeseries Forecast with log-normalized and differentiated data

i posted a similar, but more confusion question already. I have a weekly timeseries so far, which looks like this (pls ignore the red line):

My original data is (e.g.):

index  y
1    9657
2    3693
3    18063


I decided to run linear regression for this timeseries. Where I am using the previous Index as features (e.g.):

index y    y1    y2
1    9657  nan   nan
2   3693   9657  nan
3   18063  3693  9657


My idea is to only forecast the next period (one step ahead), so I always know y1,y2. My first question: 1. Is it allowed to use linear regression for this approche, cause I am using not indepented variables?

After I runned linear regression I was quite unsatisfied with the results, and as you can see from the figure it is not really a stationar timeseries, so i log my full data, basically all y values: y=log(y) and on top I changed every y (also including the y) to y= y-yprev, which should make the data even more smooth (so y = y-y1,y1=y1-y2, y2 remove). Then I used the first 70 entries to train and the last 30 to test (My full prediction line(also on the training set) looks like this:

This looks quite ok now, I would try to use it. But

2. How can i convert the y_pred back to the original data size?

3. Does it makes sense at all?