$$\frac{ 8 }{ 729 } \times { \left( \frac{ 4 }{ 18 } \right) }^{ -6 } = { \left( \frac{ 2 }{ 9 } \right) }^{ 2m-1 }$$
$m=-1$
$$\frac{8}{729}\times \left(\frac{2}{9}\right)^{-6}=\left(\frac{2}{9}\right)^{2m-1}$$
$$\frac{8}{729}\times \frac{531441}{64}=\left(\frac{2}{9}\right)^{2m-1}$$
$$\frac{729}{8}=\left(\frac{2}{9}\right)^{2m-1}$$
$$\left(\frac{2}{9}\right)^{2m-1}=\frac{729}{8}$$
$$\log(\left(\frac{2}{9}\right)^{2m-1})=\log(\frac{729}{8})$$
$$\left(2m-1\right)\log(\frac{2}{9})=\log(\frac{729}{8})$$
$$2m-1=\frac{\log(\frac{729}{8})}{\log(\frac{2}{9})}$$
$$2m-1=\log_{\frac{2}{9}}\left(\frac{729}{8}\right)$$
$$2m=-3-\left(-1\right)$$
$$m=-\frac{2}{2}$$
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