= Solution
Divide metal enrichment by gas depletion in the <closed-box model of galactic chemical evolution>:
$$
\frac{dZ}{dM_g}=-\frac{Y}{M_g}.
$$
The <logarithm> appears on integrating from $M_0$ and $Z=0$:
$$
Z=-Y\int_{M_0}^{M_g}\frac{dm}{m}=Y\ln\frac{M_0}{M_g}.
$$
With the <gas fraction of a galaxy> $\mu=M_g/M_0$, this is
$$
\boxed{Z=Y\ln(1/\mu).}
$$
It gives $Z=0$ at $\mu=1$ and increasing <gas-phase metallicity> as gas is consumed, independently of the time dependence of the <star formation rate>. For this closed initially star-free system, $M_0=M_g+M_s$, so this initial-reservoir fraction equals the present gas-to-total baryonic fraction. That equality fails in an open system.
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