what is $b(p)$ in the Almost Ideal Demand System?
In the original article by Deaton and Muellbauer (1980), An Almost Ideal Demand System, they characterize the expenditure (cost) function for the class of PIGLOG preferences as:

$$\log c(u, p) = (1-u) \log \{a(p)\} + u \log \{b(p)\}\tag{1}$$

Where they go on to describe in intuitive terms:


  " With some exceptions (see the Appendix), $u$ lies between $0$ (subsistence) and $1$ (bliss) so that the positive linearly homogeneous functions $a(p)$ and $b(p)$ can be regarded as the costs of subsistence and bliss, respectively. " (p. 313)


Some algebra gets you equations $(2)$ and $(3)$ in the paper, which are analogous forms of the equations you have above.