The ActEd notes maintain that the absolute risk aversion function:
A(w) = -U''(w)/U'(w)
Can be ascribed to the fact that "The absolute value of the certainty equivalent of a fair gamble is proportional to -U''(w)/U'(w)".
Attempting to verify this for the function U(w)=log(w) does not appear to yield that conclusion. Consider the following where w is the initial level of wealth:
A fair additive gamble with payoff x s.t.
E(Gamble) = 1/2x + 1/2(-x) = 0
Now, we have that the certainty equivalent (CE) is given by:
log(CE) = 1/2 log(w+x) + 1/2 log (w-x)
log(CE) = 1/2 log( (w+x)(w-x) )
log(CE) = log( [(w+x)(w-x)] ^ 1/2)
|c_x| = [(w+x)(w-x)] ^ 1/2 - w
-U''(w)/U'(w) = 1/w
Thus c_x does not appear to be proportional to the absolute risk aversion measure A(w).