Often to show data that is highly dispersed one will compress the data by graphing its log. One down-side of this is that it only works if the data is all-positive or all-negative (if you use \(-ln(-x)\)). If your data contains zero and/or points in both domains then the you have to do something else. Here is a simple extension that uses a linear function around zero to smoothly connect a log function and it's opposite. $$x= \begin{cases} \ln(x) & \text{if }x>e\\ x/e & \text{if }-e\leq x\leq e\\ -\ln(-x) & \text{if }x<-e \end{cases}$$ The function is log-linear-log ("trilog").
You can get a simple Stata utility -trilog- from here to make this transformation and create axis labels.
Another intuitive extension would be to shift the log and its opposite closer to zero, such as $$x= \begin{cases} \ln(x+1) & \text{if }x\geq0\\ -\ln(-x+1) & \text{if }x<0 \end{cases}$$ The downside of this is that no longer are equal proportional changes reflected as equal distance changes.
Another intuitive extension would be to shift the log and its opposite closer to zero, such as $$x= \begin{cases} \ln(x+1) & \text{if }x\geq0\\ -\ln(-x+1) & \text{if }x<0 \end{cases}$$ The downside of this is that no longer are equal proportional changes reflected as equal distance changes.
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