<p>In this article, we study the entropy dynamics of the Heisenberg spin chain with binary bond disorder. We first develop a new method that integrates the purification scheme and the time-evolving block decimation (TEBD) algorithm, named ancillary TEBD, to study the entropy dynamics of spin chains with binary bond disorder. Secondly, with the support of exact diagonalization (ED), we calculate the multifractal dimension of the eigenstates of the bond-disordered Heisenberg chain and the quench dynamics of the inverse participation ratio (IPR), finding that the dependence of the multifractal dimension on the strength of the disorder shows no critical behavior, ruling out the existence of the many-body localization transition in the system. Then, using the ED and the ancillary TEBD method, we study the entropy dynamics of the Heisenberg chain with binary bond disorder and ascertain that the quench dynamics of the entanglement entropy can be divided into four different stages, which are attributed to the competition of the spin interaction and the disorder. Our results propose a new mechanism for the generation of logarithmic scaling behavior of entropy dynamics in disordered systems. Finally, using the ancillary TEBD method, we numerically prove the existence of the transient Mpemba effect in the bond-disordered Heisenberg chain.</p>