Stationarity of cyclical economic data I'm having trouble understanding how macroeconomic or industry data could be made stationary if there's only a limited length of time series available (e.g. 2012-2019) and I have a time series that first goes up during e.g. 2012-2015 and then goes down 2016-2019 i.e. we have just one longer economic cycle in the data. This data would not be stationary even though we take log or change percentage since first the growth is positive and then negative. How do people typically tackle these situations, do you split the time series into the different periods to get a consistent slope? Many thanks for help!
First of all, just because there is an economic cycle in a data that does not imply that data is non-stationary. For example, consider the completely stationary process simulated in R below, which is based on $x_t = \phi x_{t-1} +e_t$ with $\phi=0.9$ which is stationary by construction yet it exhibits something like economic cycle.

So the first thing you should do is to make sure there actually is non-stationarity in your data and not just assume that because data exhibit some sort of cycle. Either look in the literature on the given aggregate whether in general it is considered to be integrated of some order or perform your own unit root tests (assuming there is enough observations to carry them out - you dont mention if your data is on monthly, quarterly or yearly frequency. Depending on frequency there might be plenty of data points to carry out these tests even within the given years).
Second, taking logs  is not a solution for non-stationarity even if you have long time series (and I am not even sure what you mean by change percentages). The way how you solve non-stationarity is by taking differences of your data.
For example, a simple non-stationary process is given by
$$x_t= x_{t-1} + e_t$$
The process is non stationary because $\phi=1$  which implies that present variables are fully determined by initial conditions of the system and sum of shocks. If that is the case just taking logs of variables $\ln x_t = \ln x_{t-1} +e_t$ does not solve the unit root problem as you will have the same dependence and it does not matter how long your time series is. Taking logs of your variables can be desirable for many different reasons but not related to non-stationarity per se.
What actually solves the non-stationarity/unit-root problem is to take first differences. For example, in the example above we could transform data as:
$x_t -x_{t-1}= e_t$
Which would become stationary series. You can apply the same procedure to $\ln x_t$ but what ultimately gets rid of the non-stationarity is the differencing. In a worse case scenario you could have series that is integrated of order 2 in which case you would have to make one more second difference. However, eventually taking difference of the series will always produce stationary series. According to Verbeek (2008) guide to modern macroeconomics most economic series that are non-stationary are I(1) and in some rare cases I(2), so even with extremely short time series you should be able to make it stationary.