We prove existence and uniqueness of a random field solution $(u(t,x); (t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|z| \big( \ln_+(|z|) \big)^a$ for some $a>0$. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.