Question

If (log_10(x))^2 - 4|log_10(x)| + 3 = 0 the product of roots of the equation

Answer 1

$(\log x)^2-4|\log x|+3=0\qquad(x>0)\\\\1.\ x\in(0,1\rangle\\(\log x)^2-4(-\log x)+3=0\\(\log x)^2+4\log x+3=0\\(\log x)^2+\log x+3\log x+3=0\\\log x(\log x+1)+3(\log x+1)=0\\(\log x+3)(\log x+1)=0\\\log x +3=0 \vee \log x+1=0\\\log x=-3 \vee \log x=-1\\x=\dfrac{1}{1000} \vee x=\dfrac{1}{10}\\\\\left(x=\dfrac{1}{1000} \vee x=\dfrac{1}{10}\right) \wedge x\in(0,1\rangle\\\\\underline{x=\dfrac{1}{1000} \vee x=\dfrac{1}{10}}$

$2.\ x\in(1,\infty)\\(\log x)^2-4\log x+3=0\\(\log x)^2-\log x-3\log x+3=0\\\log x(\log x-1)-3(\log x-1)=0\\(\log x-3)(\log x-1)=0\\\log x-3=0 \vee \log x-1=0\\\log x=3 \vee \log x=1\\x=1000 \vee x=10\\(x=1000 \vee x=10) \wedge x\in(1,\infty)\\\underline{x=1000 \vee x=10}\\\\x=\dfrac{1}{1000} \vee x=\dfrac{1}{10} \vee x=1000 \vee x=10$

$\boxed{\dfrac{1}{1000}\cdot\dfrac{1}{10}\cdot 10\cdot 1000=1}$