Question

Найдите корни уравнения. $x^{4lgx} =10 x^{3}$

Answer 1

ОДЗ: x>0

$x^{4*lgx}=10 x^{3} log _{x} x^{4*lgx} =log _{x}10 x^{3} 4*lgx=log _{x} 10+log _{x} x^{3}$

$4lgx= \frac{lg10}{lgx} +3. 4lgx= \frac{1}{lgx}+3 lgx=t 4t ^{2} -3t-1=0 t _{1}=- \frac{1}{4}, t x_{2} =1$

$1. t _{1} =- \frac{1}{4} , lgx=- \frac{1}{4} . x=10 ^{- \frac{1}{4} } . x _{1} = \frac{1}{ \sqrt[4]{10} }$

$2. t_{2} =1, lgx=1 x_{2} =10$