Logarithm of stock index and the non-logarithm version gives different conclusions about stationarity I am currently conducting a test of stationarity on the Shanghai Stock Exchange (SSE) index. 
Clearly when graphing both the ln(SSE) and SSE series, I see an upward trend, which should clearly indicate a violation of stationarity. I conducted an Augmented Dickey Fuller tests on both, and found that in the case of ln(SSE), I was able to reject the null hypothesis that there is a unit root in ln(SSE). This was not the case for the SSE series. 
Does this mean that when you take the logarithm of a variable, there is a slight chance of making the data stationary as what the test results suggest?
 A: Note that the null hypothesis of the ADF test is that there is a Unit root.
The alternative hypothesis of the test is that there is a "root outside the unit circle". A "root outside the Unit circle" usually infers stationarity or trend-stationarity.
Trend-stationarity CANNOT be detected with an ADF test. The ADF test does not test a null hypothesis of trend stationarity against an alternative hypothesis of strict stationarity.  Instead detrend the time series and test for stationarity in the detrended data set. 
Edit: Note that a Unit root can be defined as a shock which shifts the time series permanently to a new level. Apparently your data does not have such shocks. 
Edit2: In the following link I described cases in which an ADF-test might produce missleading results (i.e. if the time series contains components of both stationary and non-stationary time series or if the data set is two small for inference. In your case the first case might be a possibillity). Two solutions of these problems are either applying additionally for a Stationarity test such as KPSS or Leybourne-McCabe or applying a Variance ratio test .What is the difference between a stationary test and a unit root test?
