$$( \frac { 2 } { 3 } ) ^ { - 3 } \times ( \frac { 2 } { 3 } ) ^ { 8 } = ( \frac { 4 } { 9 } ) ^ { \frac { m - 1 } { 2 } }$$
$m=6$
$$\left(\frac{2}{3}\right)^{5}=\left(\frac{4}{9}\right)^{\frac{m-1}{2}}$$
$$\frac{32}{243}=\left(\frac{4}{9}\right)^{\frac{m-1}{2}}$$
$$\frac{32}{243}=\left(\frac{4}{9}\right)^{\frac{1}{2}m-\frac{1}{2}}$$
$$\left(\frac{4}{9}\right)^{\frac{1}{2}m-\frac{1}{2}}=\frac{32}{243}$$
$$\log(\left(\frac{4}{9}\right)^{\frac{1}{2}m-\frac{1}{2}})=\log(\frac{32}{243})$$
$$\left(\frac{1}{2}m-\frac{1}{2}\right)\log(\frac{4}{9})=\log(\frac{32}{243})$$
$$\frac{1}{2}m-\frac{1}{2}=\frac{\log(\frac{32}{243})}{\log(\frac{4}{9})}$$
$$\frac{1}{2}m-\frac{1}{2}=\log_{\frac{4}{9}}\left(\frac{32}{243}\right)$$
$$\frac{1}{2}m=\frac{5}{2}-\left(-\frac{1}{2}\right)$$
$$m=\frac{3}{\frac{1}{2}}$$
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