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Question Number 186996 by Mingma last updated on 12/Feb/23

Answered by aba last updated on 12/Feb/23

0<ln(3/2)<6/13

$$\mathrm{0}<\mathrm{ln}\left(\mathrm{3}/\mathrm{2}\right)<\mathrm{6}/\mathrm{13} \\ $$

Commented by MJS_new last updated on 12/Feb/23

you must show it without a calculator

$$\mathrm{you}\:\mathrm{must}\:\mathrm{show}\:\mathrm{it}\:\mathrm{without}\:\mathrm{a}\:\mathrm{calculator} \\ $$

Commented by aba last updated on 12/Feb/23

((13ln((3/2)))/6)=ln(((((3/2))^(13) ))^(1/6) )<ln(e)=1 ⇒ln((3/2))<(6/(13))

$$ \\ $$$$\frac{\mathrm{13ln}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)}{\mathrm{6}}=\mathrm{ln}\left(\sqrt[{\mathrm{6}}]{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{13}} }\right)<\mathrm{ln}\left(\mathrm{e}\right)=\mathrm{1} \\ $$$$\Rightarrow\mathrm{ln}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)<\frac{\mathrm{6}}{\mathrm{13}} \\ $$

Commented by Frix last updated on 12/Feb/23

You seriously claim to see ((((3/2))^(13) ))^(1/6) <e without calculating the value?!

$$\mathrm{You}\:\mathrm{seriously}\:\mathrm{claim}\:\mathrm{to}\:\mathrm{see}\:\sqrt[{\mathrm{6}}]{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{13}} }<\mathrm{e} \\ $$$$\mathrm{without}\:\mathrm{calculating}\:\mathrm{the}\:\mathrm{value}?! \\ $$

Commented by aba last updated on 12/Feb/23

∀n≥1 : (1+(1/n))^n ≤e ((((3/2))^(13) ))^(1/6) =((3/2))^((13)/6) =(1+(1/2))^2 ×((3/2))^(1/6) ≤((3/2))^(1/6) e ≤(1.5)^(1/6) .e <e

$$\forall\mathrm{n}\geqslant\mathrm{1}\::\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} \leqslant\mathrm{e} \\ $$$$\sqrt[{\mathrm{6}}]{\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{13}} }=\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{13}}{\mathrm{6}}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} ×\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\leqslant\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} \mathrm{e} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\leqslant\left(\mathrm{1}.\mathrm{5}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} .\mathrm{e} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:<\mathrm{e} \\ $$

Commented by Mingma last updated on 12/Feb/23

Good!

Commented by mr W last updated on 12/Feb/23

wrong sir! from ≤(1.5)^(1/6) .e you can not get <e because (1.5)^(1/6) >1.

$${wrong}\:{sir}! \\ $$$${from} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\leqslant\left(\mathrm{1}.\mathrm{5}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} .\mathrm{e} \\ $$$${you}\:{can}\:{not}\:{get} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:<\mathrm{e} \\ $$$${because}\:\left(\mathrm{1}.\mathrm{5}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} >\mathrm{1}. \\ $$

Commented by aba last updated on 12/Feb/23

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Commented by Frix last updated on 12/Feb/23

ln (3/2) <(6/(13)) ((13)/6)ln (3/2) <1 ((3/2))^((13)/6) <e ((3/2))^(1/6) ((3/2))^2 <e ((3/2))^(1/6) (9/4)<((27)/(10))<e ((3/2))^(1/6) <(6/5) (3/2)<(6^6 /5^6 ) ((3×5^6 )/(2×6^6 ))<1 (5^6 /(4×6^5 ))<1 ((15625)/(31104))<1 true

$$\mathrm{ln}\:\frac{\mathrm{3}}{\mathrm{2}}\:<\frac{\mathrm{6}}{\mathrm{13}} \\ $$$$\frac{\mathrm{13}}{\mathrm{6}}\mathrm{ln}\:\frac{\mathrm{3}}{\mathrm{2}}\:<\mathrm{1} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{13}}{\mathrm{6}}} <\mathrm{e} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} <\mathrm{e} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} \frac{\mathrm{9}}{\mathrm{4}}<\frac{\mathrm{27}}{\mathrm{10}}<\mathrm{e} \\ $$$$\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{6}}} <\frac{\mathrm{6}}{\mathrm{5}} \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}<\frac{\mathrm{6}^{\mathrm{6}} }{\mathrm{5}^{\mathrm{6}} } \\ $$$$\frac{\mathrm{3}×\mathrm{5}^{\mathrm{6}} }{\mathrm{2}×\mathrm{6}^{\mathrm{6}} }<\mathrm{1} \\ $$$$\frac{\mathrm{5}^{\mathrm{6}} }{\mathrm{4}×\mathrm{6}^{\mathrm{5}} }<\mathrm{1} \\ $$$$\frac{\mathrm{15625}}{\mathrm{31104}}<\mathrm{1}\:\mathrm{true} \\ $$

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