$$7^{2m}\div7^{-4}=7^{-6}.$$
$m=-5$
$$2401\times 7^{2m}=\frac{1}{117649}$$
$$7^{2m}=\frac{1}{282475249}$$
$$\log(7^{2m})=\log(\frac{1}{282475249})$$
$$2m\log(7)=\log(\frac{1}{282475249})$$
$$2m=\frac{\log(\frac{1}{282475249})}{\log(7)}$$
$$2m=\log_{7}\left(\frac{1}{282475249}\right)$$
$$m=-\frac{10}{2}$$
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