  Consider the stochastic reaction-diffusion equation with logarithmic
nonlinearity driven by space-time white noise: \begin{align}\label{1.a} \left\{
\begin{aligned} & \mathrm{d}u(t,x) = \frac{1}{2}\Delta u(t,x)\,\mathrm{d}t+
b(u(t,x)) \,\mathrm{d}t \nonumber\\ & ~~~~~~~~~~~~~~~~ + \sigma(u(t,x))
\,W(\mathrm{d}t,\mathrm{d}x), \ t>0, x\in I , \\ & u(0,x)=u_0(x), \quad x\in I
.\nonumber \end{aligned} \right. \end{align} When $I$ is a compact interval,
say $I=[0,1]$, the well-posedness of the above equation was established in
[DKZ] (Ann. Prob. 47:1,2019). The case where $I=\mathbb{R}$ was left open. The
essential obstacle is caused by the explosion of the supremum norm of the
solution, $\sup_{x\in\mathbb{R}}|u(t,x)|=\infty$, making the usual truncation
procedure invalid. In this paper, we prove that there exists a unique global
solution to the stochastic reaction-diffusion equation on the whole real line
$\mathbb{R}$ with logarithmic nonlinearity. Because of the nature of the
nonlinearity, to get the uniqueness, we are forced to work with the first order
moment of the solutions on the space $C_{tem}(\mathbb{R})$ with a specially
designed norm $$\sup_{t\leq T, x\in\mathbb{R}}\left(|u(t,x)|e^{-\lambda
|x|e^{\beta t}}\right),$$ where, unlike the usual norm in
$C_{tem}(\mathbb{R})$, the exponent also depends on time $t$ in a particular
way. Our approach depends heavily on the new, precise lower order moment
estimates of the stochastic convolution and a new type of Gronwall's
inequalities we obtained, which are of interest on their own right.
