autocorrelationpartial-correlationseasonalitystationaritytime series
So, I am looking my raw time series dataset, which is non stationary. I initially used the log transformation to stationarize the dataset. The plot the graph(down below). It is obvious that there is still a seasonal component to the data from the ACF plot.
I then tried to use differencing to remove the seasonal component. That resulted in me getting the plot below
I feel stuck here. How do I proceed from here ? How do I interpret the seasonality of the Log differenced plot and model the data ?
looking at plots in order to try to pigeonhole the data into a guessed arima model works well when 1: There are no outliers/pulses/level shifts, local time trends and no seasonal deterministic pulses in the data AND 2) when the arima model has constant parameters over time AND 3) when the error variance from the arima model has constant variance over time. When do these three things hold .... in most textbook data sets presenting the ease of arima modelling. When do 1 or more of the 3 not hold .... in every real world data set that I have ever seen . The simple answer to your question requires access to the original facts ( the historical data ) not the secondary descriptive information in your plots. But this is just my opinion!
EDITED AFTER RECEIPT OF DATA:
I was on a Greek vacation (actually doing something other than time series analysis) and was unable to analyse the SUICIDE DATA but in conjunction with this post. It is now fitting and right that I submit an analysis to follow up/prove by example my comments about multi-stage model identification strategies and the failings of simple visual analysis of simple correlation plots as "the proof is in the pudding".
Here is the ACF of the original data 
The PACF of the original series 
. AUTOBOX http://www.autobox.com/cms/ a piece of software that I have helped developed uses heuristics to identify a starting model In this case the initially identified model was found to be 
. Diagnostic checking of the residuals from this model suggested some model augmentation using a level shift, pulses and a seasonal pulse Note that the Level Shift is detected at or about period 164 which is nearly identical to an earlier conclusion about period 176 from @forecaster. All roads do not lead to Rome but some can get you close ! 
. Testing for parameter constancy rejected parameter changes over time . Checking for deterministic changes in the error variance concluded that no deterministic changes were detected in the error variance.
. The Box-Cox test for the need for a power transform was positive with the conclusion that a logarithmic transform was necessary.
. The final model is here 
. The residuals from the final model appear to be free of any autocorrelation 
. The plot of the final models residuals appears to be free of any Gaussian Violations 
. The plot of Actual/Fit/Forecasts is here 
with forecasts here 
At a glance your data appears to potentially have seasonality. It is another question as to the kind of seasonality structure that is appropriate.
Data like yours can arise from a model that might include seasonal ARIMA structure or seasonal dummies ... the eye can't easily sort this out: Only the data knows for sure!
What I am suggesting is that the statistical characteristics of alternative models should be employed to evaluate these two alternative approaches. This can be done by evaluating tentative alternatives including seasonal dummies that might even change over time – e.g., a "January effect" that changed at a particular point in time.
Furthermore it is always possible that both forms of seasonality are needed to render an error process free of structure (i.e., gaussian). If you wish you can post your data and I will try to help further possibly just using visual methods. If your data is confidential simply scale it.
EDITED TO ANSWER OP'S QUESTION AS TO WHAT PLOT WOULD SUGGEST THE FORM OF THE SEASONALITY:
In this case the plot that would tell you the kind of seasonality that is present would be the ACF of the first differences.
Best Answer
As you've rightly pointed out, the ACF in the first image clearly shows an annual seasonal trend wrt. peaks at yearly lag at about 12, 24, etc. The log-transformed series represents the series scaled to a logarithmic scale. This represents the size of the seasonal fluctuations and random fluctuations in the log-transformed time series which seem to be roughly constant over the yearly seasonal fluctuation and does not seem to depend on the level of the time series.
Since, we observe annual seasonality, the most appropriate $d$-th order differencing for this data set seems to be the $12$-th order differencing. Then, the log-transformed series is expected to represent a randomly fluctuated log-series. The elimination of the annual cycle seems about right.