Question

Неравенство с логарифмом - решите пожалуйста подробно

Answer 1

Ответ:

$x^2\cdot log_{25}\, x\geq log_{25}\, x^3+x\cdot log_5\, x\ \ ,\ \ \ \ ODZ:\ x>0\ ,\ x\in (\ 0\ ;+\infty \, )\\\\x^2\cdot \dfrac{1}{2}\cdot log_5\, x-\dfrac{3}{2}\cdot log_5\, x-x\cdot log_5\, x\geq 0\\\\log_5\, x\cdot (0,5x^2-x-1,5)\geq 0$

$a)\ \ \left\{\begin{array}{l}log_5\, x\geq 0\\0,5x^2-x-1,5\geq 0\end{array}\right\ \ \left\{\begin{array}{l}x\geq 1\\(x+1)(x-3)\geq 0\end{array}\right\\\\\\\left\{\begin{array}{l}x\geq 1\\x\in (-\infty \, ;\, -1\, ]\cup [\ 3\ ;+\infty \, )\end{array}\right\ \ \Rightarrow \ \ \ x\in [\ 3\ ;+\infty )$

$b)\ \ \left\{\begin{array}{l}log_5\, x\leq 0\\0,5x^2-x-1,5\leq 0\end{array}\right\ \ \left\{\begin{array}{l}0<x\leq 1\\(x+1)(x-3)\leq 0\end{array}\right\\\\\\\left\{\begin{array}{l}0<x\leq 1\\x\in [ -1\, ;\ 3\ ]\end{array}\right\ \ \Rightarrow \ \ \ x\in (\ 0\ ;\ 1\ ]\\\\\\Otvet:\ x\in (\ 0\ ;\ 1\ ]\cup [\ 3\ ;\ +\infty \, )\ .$