In this work, static traversable wormholes are investigated in $$R^2$$ $R^2$ gravity with logarithmic trace term T, where R denotes the Ricci scalar, and $$T=-\rho +p_r+2p_t>0$$ $T=-\rho +p_r+2p_t>0$ , the trace of the energy momentum tensor. The connection between energy density of the matter component and the Ricci scalar is taken into account. Exact wormhole solutions are determined for three different novel forms of energy density: $$\rho =\alpha _1 R+\beta _1 R^{'}e^{\xi _1 R}$$ $\rho ={\alpha }_{1}R+{\beta }_1{R}^{{}^{\prime }}{e}^{{\xi }_{1}R}$ , $$\rho =\alpha _2 R e^{\xi _2 R}$$ $\rho ={\alpha }_{2}R{e}^{{\xi }_{2}R}$ and $$\rho =\alpha _3 R^2+\beta _2 R^{'} e^{\xi _3 R^{'}}$$ $\rho ={\alpha }_3{R}^2+{\beta }_2{R}^{{}^{\prime }}{e}^{{\xi }_3{R}^{{}^{\prime }}}$ , where prime denotes derivative with respect to r. The parameters $$\alpha _1$$ ${\alpha }_1$ , $$\beta _1$$ ${\beta }_1$ , $$\xi _1$$ ${\xi }_1$ , $$\alpha _2$$ ${\alpha }_2$ , $$\xi _2$$ ${\xi }_2$ , $$\alpha _3$$ ${\alpha }_3$ , $$\xi _3$$ ${\xi }_3$ and $$\beta _2$$ ${\beta }_2$ play an important role for the absence of exotic matter inside the wormhole geometry. The parameter space is separated into numerous regions where the energy conditions are obeyed.