Question Number 207590 by hardmath last updated on 19/May/24
$$\mathrm{x}^{\:\boldsymbol{\mathrm{lg}}\:\boldsymbol{\mathrm{x}}} \:\:<\:\:\mathrm{10}^{\mathrm{4}} \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{roots}? \\ $$
Commented by hardmath last updated on 19/May/24
$$\mathrm{professor},\:\mathrm{integer}\:\mathrm{roots} \\ $$
Answered by mr W last updated on 19/May/24
$${x}^{\mathrm{log}\:{x}} <\mathrm{10}^{\mathrm{4}} \\ $$$$\left(\mathrm{log}\:{x}\right)\left(\mathrm{log}\:{x}\right)<\mathrm{4} \\ $$$$\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} <\mathrm{4} \\ $$$$−\mathrm{2}<\mathrm{log}\:{x}<\mathrm{2} \\ $$$$\mathrm{10}^{−\mathrm{2}} <{x}<\mathrm{10}^{\mathrm{2}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{100}}<{x}<\mathrm{100} \\ $$$${for}\:{x}\in{Z}: \\ $$$$\mathrm{1}\leqslant{x}\leqslant\mathrm{99}\:\Rightarrow\mathrm{99}\:{roots}! \\ $$
Commented by hardmath last updated on 20/May/24
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{professor} \\ $$