In this paper, we prove that there exists a unique global solution to the stochastic reaction-diffusion equation with logarithmic nonlinearity driven by space-time white noise on the whole real line $\mathbb R$. The essential obstacle is caused by the explosion of the supremum norm of the solution, making the usual truncation procedure invalid. Our approach depends heavily on a specially designed norm of the space $C_{tem}(\mathbb R)$, and new, precise lower order moment estimates of the stochastic convolution and a new type of Gronwall's inequalities we obtained.