Clarifying the dynamic evolution characteristics and driving mechanism of the global log trade network (GLTN) can provide references for the trade decision-making of various countries. Based on the stochastic actor-oriented model, this paper analyzed the GLTN’s dynamic evolution from 2010 to 2019 and paid more attention to the effect of forest certification on the dynamic evolution of the GLTN. Results indicate that from 2010 to 2019, many changes have occurred in the network. The change rate in the 2010–2015 period is faster than that in the 2015–2019 period. Countries tend to form reciprocity, three-cycles, and transitive trade ties over time. A country with more certified forest area tends to form new log export ties with the other countries. The trade imbalance ratio (TII) has a significant negative mediating effect on the evolutionary relationships between the certified forest area and the log trade network dynamic. Net exporters of log tend to avoid establishing export ties with countries with more certified forest areas over time. Countries with similar cultural backgrounds are easier to establish log trade ties with, while countries with farther geographical distances tend to avoid establishing trade ties over time. A free trade agreement (FTA) has a significant positive impact on the formation of log trade ties among nations. Our findings shed new light on the dynamic evolution mechanism of the global log trade network and offer implications for future trade and forest conservation policy design.