+118608 The question here was not posted properly so I could not answer on the original question. 
I think it was because you did not use plain words in your heading.
I have posted my answer here instead .
$${{\mathtt{2}}}^{{\mathtt{2\,009}}}{\mathtt{\,-\,}}{{\mathtt{2}}}^{{\mathtt{2\,007}}}$$
$${{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2\,007}}}{\mathtt{\,-\,}}{{\mathtt{2}}}^{{\mathtt{2\,007}}}$$
$${{\mathtt{2}}}^{{\mathtt{2\,007}}}{\mathtt{\,\times\,}}\left({\mathtt{4}}{\mathtt{\,-\,}}{\mathtt{1}}\right)$$
$${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2\,007}}}$$
let y= $${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2\,007}}}$$
$$\\\frac{y}{3}=2^{2007}\\
log(\frac{y}{3})=log(2^{2007})\\
log(y)-log(3)=2007*log(2)\\
log(y)=2007*log(2)+log(3)$$
$${\mathtt{2\,007}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{2}}\right){\mathtt{\,\small\textbf+\,}}{log}_{10}\left({\mathtt{3}}\right) = {\mathtt{604.644\: \!322\: \!552\: \!329\: \!930\: \!8}}$$
$$\\y=10^{604.6443225523299308}\\\\
y=10^{604}*10^{0.6443225523299308}\\\\
y=10^{0.6443225523299308}*10^{604}$$
$${{\mathtt{10}}}^{{\mathtt{0.644\: \!322\: \!552\: \!329\: \!930\: \!8}}} = {\mathtt{4.408\: \!821\: \!869\: \!853\: \!234\: \!5}}$$
$$\\y=4.40088218698532345\times 10^{604}\\\\
so\\\\
2^{2009}-2^{2007}=4.40088218698532345\times 10^{604}$$
That is assumiong that I did not make any stupid mistakes :/
