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

However,

-U''(w)/U'(w) = 1/w

Thus c_x does not appear to be proportional to the absolute risk aversion measure A(w).