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How do you graph ##ln(abs(x))##?
The typical graph of just ##ln(x)## is
graph{ln(x) [-10, 10, -5, 5]}
Notice the restriction. In ##ln(x)##, ##x>0##. That is, negative numbers are not in the domain of a logarithmic function.
However, in ##ln(abs(x))##, negative numbers are made positive.
For example, both ##e^2## and ##-e^2##, when plugged into ##ln(abs(x))##, result in ##ln(e^2)=2##.
In effect, adding the absolute value makes both the positive and negative realms available for the natural logarithm, in effect reflecting the graph over the ##y##-axis, while retaining itself on the positive side:
