We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings
J smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths
l is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length
n increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
We investigate the magnetic susceptibility $\ensuremath{\chi}(T)$ of quantum spin chains of
N=1280 spins with power-law long-range antiferromagnetic couplings as a function of their spatial decay exponent $\ensuremath{\alpha}$ and cutoff length $\ensuremath{\xi}$. The calculations are based on the strong disorder renormalization method which is used to obtain the temperature dependence of $\ensuremath{\chi}(T)$ and distribution functions of couplings at each renormalization step. For the case with only algebraic decay $(\ensuremath{\xi}=\ensuremath{\infty})$ we find a crossover at ${\ensuremath{\alpha}}^{*}=1.066$ between a phase with a divergent low-temperature susceptibility $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ to a phase with a vanishing $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$. For finite cutoff lengths $\ensuremath{\xi}$, this crossover occurs at a smaller ${\ensuremath{\alpha}}^{*}(\ensuremath{\xi})$. Additionally, we study the localization of spin excitations for $\ensuremath{\xi}=\ensuremath{\infty}$ by evaluating the distribution function of excitation energies, and we find a delocalization transition that coincides with the opening of the pseudogap at ${\ensuremath{\alpha}}_{c}={\ensuremath{\alpha}}^{*}$.