Assess stationarity of a growth variable

Dear Matlab community,

I am trying to assess a time series variable for stationarity. To give you an idea what I am doing: The variable is "industrial production index" (INDPRO), which measures industrial output in the U.S.

As it may be a growth variable, I first took the logarithm:

logINDPRO = log(INDPRO);

I then took the first difference, using

DlogINDPRO = diff(logINDPRO);

The augmented dickey fuller test gives =1 as an output, so does the KPSS test. Thus, I took up to three differences, but it seems there is not really an improvement:

Here a time series plot and a correlogram for illustration:

It appears that there is still autocorrelation (or perhaps a trend?) in my process. I am not sure what to do next: Should I keep taking differences, or will I probably not be able to pass the text this way anyway?

I suspect that the shocks in the process cause failing the tests.

Is there probably another test which is more appropriate?

I believe that KPSStest and ADFtest should give me a test output of '0' in order to pass?

Thank you very much for your help, it is much appreciated. I am spending quite some time on this now.

Greeting from Australia,

Chris