  In this article, we establish a characterization of the set
$M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besov
spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness
$b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications of
these two characterizations, we clarify whether or not the three concrete
examples, namely characteristic functions of open sets, continuous functions
defined by differences, and the functions $e^{ik\cdot x}$ with
$k\in\mathbb{Z}^n$ and $x\in\mathbb{R}^n$, are pointwise multipliers of
$B^{0,b}_{1,\infty}(\mathbb{R}^n)$ and $B^{0,b}_{\infty,\infty}(\mathbb{R}^n)$,
respectively; furthermore, we obtain the explicit estimates of $\|e^{ik \cdot
x}\|_{M(B^{0,b}_{1,\infty}(\mathbb{R}^n))}$ and $\|e^{ik \cdot
x}\|_{M(B^{0,b}_{\infty,\infty}(\mathbb{R}^n))}$. In the case that
$p\in(1,\infty)$, we give some sufficient conditions and some necessary
conditions of the pointwise multipliers of $B^{0,b}_{p,\infty}(\mathbb{R}^n)$
and a complete characterization of $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ is
still open. However, via a different method, we are still able to accurately
calculate $\|e^{ik \cdot x}\|_{M(B^{0,b}_{p,\infty}(\mathbb{R}^n))}$,
$k\in\mathbb{Z}^n$, in this situation. The novelty of this article is that most
of the proofs are constructive and these constructions strongly depend on the
logarithmic structure of Besov spaces under consideration.
