$$( \frac { 3 } { 2 } ) ^ { - 17 } \div ( \frac { 3 } { 2 } ) = ( \frac { 3 } { 2 } ) ^ { 2 m + 1 }$$
$m = -\frac{19}{2} = -9\frac{1}{2} = -9.5$
$$\frac{1}{\left(\frac{3}{2}\right)^{18}}=\left(\frac{3}{2}\right)^{2m+1}$$
$$\frac{1}{\frac{387420489}{262144}}=\left(\frac{3}{2}\right)^{2m+1}$$
$$1\times \frac{262144}{387420489}=\left(\frac{3}{2}\right)^{2m+1}$$
$$\frac{262144}{387420489}=\left(\frac{3}{2}\right)^{2m+1}$$
$$\left(\frac{3}{2}\right)^{2m+1}=\frac{262144}{387420489}$$
$$\log(\left(\frac{3}{2}\right)^{2m+1})=\log(\frac{262144}{387420489})$$
$$\left(2m+1\right)\log(\frac{3}{2})=\log(\frac{262144}{387420489})$$
$$2m+1=\frac{\log(\frac{262144}{387420489})}{\log(\frac{3}{2})}$$
$$2m+1=\log_{\frac{3}{2}}\left(\frac{262144}{387420489}\right)$$
$$2m=-18-1$$
$$m=-\frac{19}{2}$$
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