Answer
If b=0, take, for example $x_{1}=1$ and $x_{2}=2.$
They are obviously different, $x_{1}\neq x_{2}$, but
$b^{x_{1}}=0^{1}=0,\qquad b^{x_{2}}=0^{2}=0$, meaning that $b^{x_{1}}=b^{x_{2}}$.
If b=1, take, for example $x_{1}=1$ and $x_{2}=2.$
They are obviously different, $x_{1}\neq x_{2}$, but
$b^{x_{1}}=1^{1}=1,\qquad b^{x_{2}}=1^{2}=1$, meaning that $b^{x_{1}}=b^{x_{2}}$.
The equivalence fails in both cases.
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