a.
$
\eqalign{
& {}^3\log \left( {2x^2 - 3} \right) = 6 \cr
& 2x^2 - 3 = 3^6 \cr
& 2x^2 - 3 = 729 \cr
& 2x^2 = 732 \cr
& x^2 = 366 \cr
& x = - \sqrt {366} \,\,of\,\,x = \sqrt {366} \cr}
$
b.
$
\eqalign{
& {}^{\frac{1}
{2}}\log \left( {\frac{1}
{{4x}}} \right) = 4 \cr
& \frac{1}
{{4x}} = \left( {\frac{1}
{2}} \right)^4 \cr
& \frac{1}
{{4x}} = \frac{1}
{{16}} \cr
& 4x = 16 \cr
& x = 4 \cr}
$
c.
$
\eqalign{
& {}^2\log \left( {4 - 30x^2 } \right) = - 2 \cr
& 4 - 30x^2 = 2^{ - 2} \cr
& 4 - 30x^2 = \frac{1}
{4} \cr
& 16 - 120x^2 = 1 \cr
& 120x^2 = 15 \cr
& x^2 = \frac{1}
{8} \cr
& x = - \sqrt {\frac{1}
{8}} \,\,of\,\,x = \sqrt {\frac{1}
{8}} \cr
& x = - \frac{1}
{4}\sqrt 2 \,\,of\,\,x = \frac{1}
{4}\sqrt 2 \cr}
$