Question Number 94298 by peter frank last updated on 17/May/20
Answered by Ar Brandon last updated on 18/May/20
$$\mathrm{log}_{\mathrm{2}} \left(\mathrm{1}×\mathrm{2}×\mathrm{3}×...×\mathrm{n}\right)=\mathrm{1994} \\ $$$$\mathrm{log}_{\mathrm{2}} \left(\mathrm{n}!\right)=\mathrm{1994}\Rightarrow\mathrm{n}!=\mathrm{2}^{\mathrm{1994}} \\ $$$$\mathrm{n}\approx\mathrm{295} \\ $$
Commented by peter frank last updated on 18/May/20
$${how}\:\:{n}=\mathrm{295} \\ $$$${n}!=\mathrm{2}^{\mathrm{1994}} ? \\ $$$$ \\ $$
Commented by Ar Brandon last updated on 18/May/20
$$\mathrm{log}\left(\mathrm{a}\right)+\mathrm{log}\left(\mathrm{b}\right)+\mathrm{log}\left(\mathrm{c}\right) \\ $$$$=\mathrm{log}\left(\mathrm{abc}\right) \\ $$$$\Rightarrow\mathrm{log}_{\mathrm{2}} \left(\mathrm{1}\right)+\mathrm{log}_{\mathrm{2}} \left(\mathrm{2}\right)+\mathrm{log}_{\mathrm{2}} \left(\mathrm{3}\right)+...+\mathrm{log}_{\mathrm{2}} \left(\mathrm{n}\right)=\mathrm{log}_{\mathrm{2}} \left(\mathrm{n}!\right)=\mathrm{1994} \\ $$$$\mathrm{n}!=\mathrm{2}^{\mathrm{1994}} \\ $$
Commented by peter frank last updated on 18/May/20
$${how}\:{n}=\mathrm{295}? \\ $$
Commented by Ar Brandon last updated on 18/May/20
$$\mathrm{It}'\mathrm{s}\:\mathrm{just}\:\mathrm{an}\:\mathrm{approximation}\:\mathrm{by}\:\mathrm{the}\:\mathrm{way} \\ $$$$\mathrm{there}\:\mathrm{exist}\:\mathrm{no}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{n}!=\mathrm{2}^{\mathrm{1994}} \\ $$
Commented by peter frank last updated on 18/May/20
$${thank}\:{you} \\ $$
Commented by Ar Brandon last updated on 18/May/20
Alright. Perhaps the others will bring forth better ideas. ��