  Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be
the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence
$(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$
x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a
$q$-expansion of $x$. In this paper we investigate the local dimension of
$\mathcal U(x)$ and prove a `variation principle' for unique non-integer base
expansions. We also determine the critical values of $\mathcal U(x)$ such that
when $x$ passes the first critical value the set $\mathcal U(x)$ changes from a
set with positive Hausdorff dimension to a countable set, and when $x$ passes
the second critical value the set $\mathcal U(x)$ changes from an infinite set
to a singleton. Denote by $\mathbf U(x)$ the set of all unique $q$-expansions
of $x$ for $q\in\mathcal U(x)$. We give the Hausdorff dimension of $\mathbf
U(x)$ and show that the dimensional function $x\mapsto\dim_H\mathbf U(x)$ is a
non-increasing Devil's staircase. Finally, we investigate the topological
structure of $\mathcal U(x)$. In contrast with $x=1$ that $\mathcal U(1)$ has
no isolated points, we prove that for typical $x>0$ the set $\mathcal U(x)$
contains isolated points.
