<p>We analyze all individual cosmic strings of various lengths in a large ensemble of the global cosmic string networks in the post-inflationary scenario, obtained from numerical simulations on a discrete lattice with <i>N</i> ³ = 4096³. A strong evidence for a logarithmically growing spectral index of the string power spectrum during the evolution is newly reported as our main result. The logarithmic scaling is checked against two different approaches for generating initial random field configurations, namely fat-string type and thermal phase transition. We derive the analytic relation between two power spectra of cosmic strings and axions which should be valid under some assumptions, and the validity of those assumptions is discussed. We argue that our analytic result strongly supports the correlated spectra of cosmic strings and axions. Additionally, we initiate the statistical analysis of the causal dynamics of the cosmic strings.</p>