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Some of the reasons are convention (and the reason $1$ is the convention is that after substituting $x_i = e^{y_i}$ and taking logs of both sides, we get a convex program where all right-hand sides are $0$). But the primary practical reason to put the constraints in a standard form - that is, $g_i(\mathbf x) \le 1$ - is that then the dual of a geometric ...
$\log 1 = 0$ could explain that: take $\log$ of both sides. Edit: I have now looked and found this in "A tutorial on geometric programming", Stephen Boyd · Seung-Jean Kim · Lieven Vandenberghe · Arash Hassibi. Look on p. 73 equation (6), and the paragraph which precedes it: The conversion of a GP to a convex problem is based on a logarithmic change of ...