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Question Number 189140 by 073 last updated on 12/Mar/23

Answered by Ar Brandon last updated on 12/Mar/23

$$x>0\wedge x⩾1\Rightarrow x⩾1$$$$4-\ln x⩾3\sqrt{\ln x},t^2=\ln x$$$$4-t^2⩾3t\Rightarrow t^2+3t-4⩽0$$$$(t+4)(t-1)⩽0\Rightarrow -4⩽t⩽1$$$$0⩽t^2⩽1\Rightarrow 0⩽\ln x⩽1$$$$1⩽x⩽e\Rightarrow x\in [1,e]$$$$\Rightarrow b-a=e-1$$

Commented by manxsol last updated on 12/Mar/23

$$Sr.ArBrandon,contodorespeto.$$$$exercise=log\left(x\right)base10$$$$youexerciseln\left(x\right)basee$$

Commented by Ar Brandon last updated on 12/Mar/23

$$\mathrm{Hola}\mathrm{se}\stackrel{}{\mathrm{n}or}!$$$$\log \left(x\right)\mathrm{is}\mathrm{not}\mathrm{necessarily}\mathrm{to}\mathrm{base}10.\mathrm{It}\mathrm{all}\mathrm{depends}\mathrm{on}\mathrm{your}\mathrm{education}$$$$\mathrm{system}.\mathrm{In}\mathrm{some}\mathrm{countries}\log x\mathrm{implies}\mathrm{to}\mathrm{the}\mathrm{base}10\mathrm{while}\mathrm{in}\mathrm{others}\mathrm{it}$$$$\mathrm{is}\mathrm{to}\mathrm{the}\mathrm{base}e.\mathrm{By}\mathrm{the}\mathrm{way}\mathrm{in}\mathrm{most}\mathrm{programming}\mathrm{languages}\mathrm{like}\mathrm{the}\mathrm{C}$$$$\mathrm{programming}\mathrm{language}\mathrm{the}\mathrm{function}\log \left(x\right)\mathrm{is}\mathrm{to}\mathrm{the}\mathrm{base}e.\mathrm{And}\mathrm{to}\mathrm{use}$$$$\mathrm{the}\mathrm{base}10\mathrm{instead}\mathrm{we}\mathrm{must}\mathrm{precise}\log 10\left(x\right).$$$$\mathrm{Whatever}\mathrm{be}\mathrm{the}\mathrm{case}\mathrm{the}\mathrm{logic}\mathrm{applied}\mathrm{in}\mathrm{the}\mathrm{above}\mathrm{solution}\mathrm{remains}\mathrm{same}.$$$$\mathrm{Gracias}\mathrm{por}\mathrm{prestar}\mathrm{atenci}\acute{\mathrm{o}n}\mathrm{a}\mathrm{mi}\mathrm{soluci}\acute{\mathrm{o}n}.$$

Answered by mehdee42 last updated on 13/Mar/23

$$A)x>0$$$$B)logx⩾0\Rightarrow x⩾10$$$$C)(\sqrt{logx}+4)(\sqrt{logx}-1)⩽0\Rightarrow$$$$\sqrt{logx}-1⩽0\Rightarrow x⩽10$$$$\cap A,B,C=[1,10]\Rightarrow b-a=9$$

Commented by manolex last updated on 13/Mar/23

$$B)x\gg 1$$

Commented by 073 last updated on 13/Mar/23

$$\mathrm{thanks}\mathrm{alot}$$$$$$

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