e^7>(3^7/10)^7>1046>2^10 log2<7/10
7^5=16807 2^14=16384 7^5/2^14=1+423/16384
log(7^5/2^14)=log(1+423/16384)>423/16384-(423/16384)^2/2>1/40
5log7>14log2+1/40=14log2+7/10/28>14log2+log2/28=393/28*log2
log7/log8>(393/28)/3/5=131/140>√(7/8)