Answer
$x=\pm 1$
Work Step by Step
The base $b$ logarithm of $x,\log_{b}x,$
is the power to which we need to raise $b$ in order to get $x$ .
Symbolically, $\quad \log_{b}x=y\quad $ means $\quad b^{y}=x$
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$2^{2x^{2}-1}=2\quad $ means $\quad 2x^{2}-1=\log_{2}2,\qquad$...and, $\log_{2}2=1$
$2x^{2}-1=1$
$2x^{2}=2$
$x^{2}=1$
$x=\pm 1$
