Mass per logarithmic size bin ($m_{\log }\propto D^{4-\alpha }$)

For equal-density spherical bodies with differential number distribution $n(D)\propto D^{-\alpha }$, a fixed logarithmic bin has width $dD=Dd\log D$. Multiplying the body mass proportional to $D^3$ by $n(D)dD$ gives $m_{\log }\propto D^{4-\alpha }$. Distinguishing differential-number and logarithmic-mass slopes is essential in collisional cascade plots.