In this work, a new model of double-Logarithmic electrodynamics has been minimally coupled to Einstein-massive gravity and the nonlinearly charged black holes in the presence of quintessential cosmic dark energy and string cloud background are investigated. In this set up, new magnetically charged black hole solution has been derived. Thermodynamics of these black holes are also analyzed and different thermodynamic quantities are computed. Basic thermodynamic quantities such as heat capacity and Hawking temperature have been plotted and the regions corresponding to thermodynamic stability and instability are pointed out. The generalized first law of thermodynamics and the associated Smarr’s relation have also been written down in this context of massive gravity and matter sources. At last, the nonlinearly electrically charged black holes are also briefly studied.

中文翻译：

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被弦云包围的非线性带电黑洞和爱因斯坦大质量引力的精髓

在这项工作中，一种新的双对数电动力学模型与爱因斯坦质量引力最小耦合，并研究了存在典型宇宙暗能量和弦云背景下的非线性带电黑洞。在这个装置中，新的带磁黑洞解决方案已经被推导出来。还分析了这些黑洞的热力学并计算了不同的热力学量。绘制了热容量和霍金温度等基本热力学量，并指出了热力学稳定性和不稳定性对应的区域。广义的热力学第一定律和相关的 Smarr 关系也已在大质量重力和物质源的背景下写下来。最后，