Question Number 190395 by mr W last updated on 02/Apr/23
$${which}\:{is}\:{larger}, \\ $$$$\mathrm{2}^{\mathrm{234}} \:{or}\:\mathrm{5}^{\mathrm{100}} \:? \\ $$
Answered by aba last updated on 02/Apr/23
$$\mathrm{2}^{\mathrm{234}} >\mathrm{5}^{\mathrm{100}} \\ $$
Commented by Frix last updated on 02/Apr/23
$$\mathrm{Prove}\:\mathrm{it}? \\ $$
Answered by nikif99 last updated on 02/Apr/23
$$\mathrm{234}\:\mathrm{log}\:\mathrm{2}\lessgtr\mathrm{100}\:\mathrm{log}\:\mathrm{5}=\mathrm{100}\left(\mathrm{log}\:\left(\frac{\mathrm{10}}{\mathrm{2}}\right)\right)= \\ $$$$\mathrm{100}\left(\mathrm{log}\:\mathrm{10}−\mathrm{log}\:\mathrm{2}\right)=\mathrm{100}−\mathrm{100}\:\mathrm{log}\:\mathrm{2}\:\Rightarrow \\ $$$$\mathrm{234}\:\mathrm{log}\:\mathrm{2}\lessgtr\mathrm{100}−\mathrm{100}\:\mathrm{log}\:\mathrm{2}\:\Rightarrow \\ $$$$\mathrm{334}\:\mathrm{log}\:\mathrm{2}\lessgtr\mathrm{100}\:\Rightarrow\mathrm{334}×\mathrm{0}.\mathrm{30103}>\mathrm{100} \\ $$
Commented by Frix last updated on 02/Apr/23
$$\mathrm{Prove}\:\mathrm{must}\:\mathrm{be}\:\mathrm{without}\:\mathrm{calculation}. \\ $$$$\mathrm{Otherwise}\:\mathrm{you}\:\mathrm{can}\:\mathrm{just}\:\mathrm{use}\:\mathrm{a}\:\mathrm{calculator}. \\ $$
Answered by mahdipoor last updated on 02/Apr/23
$$\frac{\mathrm{5}^{\mathrm{100}} }{\mathrm{2}^{\mathrm{234}} }=\frac{\mathrm{5}×\mathrm{125}^{\mathrm{33}} }{\mathrm{8}×\mathrm{128}^{\mathrm{33}} }<\mathrm{1}\Rightarrow\mathrm{5}^{\mathrm{100}} <\mathrm{2}^{\mathrm{234}} \\ $$
Commented by Frix last updated on 02/Apr/23
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Commented by mr W last updated on 02/Apr/23
$${nice}! \\ $$