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Extensive modeling of the seeding of plasma-based soft X-ray lasers is reported in this article. Seminal experiments on amplification in plasmas created from solids have been studied in detail and explained. Using a transient collisional excitation scheme, we show that a 18 µJ, 80 fs fully coherent pulse is achievable by using plasmas pumped by a compact 10 Hz laser. We demonstrate that direct seeding of plasmas created by nanosecond lasers is not efficient. Therefore, we propose and fully study the transposition to soft X-rays of the Chirped Pulse Amplification (CPA) technique. Soft X-ray pulses with energy of 6 mJ and 200 fs duration are reachable by seeding plasmas pumped by compact 100 J, sub-ns, 1 shot/min lasers. These soft X-ray lasers would reach GW power, corresponding to an increase of 100 times as compared to the highest peak power achievable nowadays in the soft X-ray region (30 eV-1 keV). X-ray CPA is opening new horizon for soft x-ray ultra-intense sources. [ABSTRACT FROM AUTHOR]