Algorithm for solving fractional inequalities with logarithms. Preparation for the Unified State Exam. Solving logarithmic and exponential inequalities using the rationalization method

An inequality is called logarithmic if it contains logarithmic function.

Solution methods logarithmic inequalities no different from , except for two things.

Firstly, when moving from the logarithmic inequality to the inequality of sublogarithmic functions, one should follow the sign of the resulting inequality. It obeys the following rule.

If the base of the logarithmic function is greater than $1$, then when moving from the logarithmic inequality to the inequality of sublogarithmic functions, the sign of the inequality is preserved, but if it is less than $1$, then it changes to the opposite.

Secondly, the solution to any inequality is an interval, and, therefore, at the end of solving the inequality of sublogarithmic functions it is necessary to create a system of two inequalities: the first inequality of this system will be the inequality of sublogarithmic functions, and the second will be the interval of the domain of definition of the logarithmic functions included in the logarithmic inequality.

Practice.

Let's solve the inequalities:

1. $\log_(2)((x+3)) \geq 3.$

$D(y): \x+3>0.$

$x \in (-3;+\infty)$

The base of the logarithm is $2>1$, so the sign does not change. Using the definition of logarithm, we get:

$x+3 \geq 2^(3),$

$x \in )