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Compare-it-a-log-3-4-b-log-5-6-c-log-6-2-




Question Number 208463 by hardmath last updated on 16/Jun/24
Compare it:  a = log_3  4  b = log_5  6  c = log_6  2
$$\mathrm{Compare}\:\mathrm{it}: \\ $$$$\mathrm{a}\:=\:\mathrm{log}_{\mathrm{3}} \:\mathrm{4} \\ $$$$\mathrm{b}\:=\:\mathrm{log}_{\mathrm{5}} \:\mathrm{6} \\ $$$$\mathrm{c}\:=\:\mathrm{log}_{\mathrm{6}} \:\mathrm{2} \\ $$
Answered by Frix last updated on 16/Jun/24
3^a =4 ⇒ a>1     3^a =3+1                                                     ⇒ a>b  5^b =6 ⇒ b>1      5^b =5+1    6^c =2 ⇒ c<1    ⇒ a>b>c
$$\mathrm{3}^{{a}} =\mathrm{4}\:\Rightarrow\:{a}>\mathrm{1}\:\:\:\:\:\mathrm{3}^{{a}} =\mathrm{3}+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:{a}>{b} \\ $$$$\mathrm{5}^{{b}} =\mathrm{6}\:\Rightarrow\:{b}>\mathrm{1}\:\:\:\:\:\:\mathrm{5}^{{b}} =\mathrm{5}+\mathrm{1} \\ $$$$ \\ $$$$\mathrm{6}^{{c}} =\mathrm{2}\:\Rightarrow\:{c}<\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow\:{a}>{b}>{c} \\ $$
Commented by hardmath last updated on 16/Jun/24
thankyou dear professor
$$\mathrm{thankyou}\:\mathrm{dear}\:\mathrm{professor} \\ $$

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