Definition of logarithm The easiest way to write it mathematically is:
The definition of logarithm can be written in another way:
Pay attention to the restrictions that are imposed on the base of the logarithm ( a) and to the sublogarithmic expression ( x). In the future, these conditions will turn into important restrictions for OD, which will need to be taken into account when solving any equation with logarithms. So, now, in addition to the standard conditions leading to restrictions on ODZ (positivity of expressions under the roots of even powers, non-equal denominator to zero, etc.), the following conditions must also be taken into account:
- Sublogarithmic expression can only be positive.
- The base of the logarithm can only be positive and not equal to one.
Note that neither the base of the logarithm nor the sublogarithmic expression can be equal to zero. Please also note that the logarithm value itself can take on all possible values, i.e. The logarithm can be positive, negative or zero. Logarithms have many different properties that follow from the properties of powers and the definition of a logarithm. Let's list them. So, the properties of logarithms:
Logarithm of the product:
Logarithm of a fraction:
Taking the degree out of the logarithm sign:
Pay special attention close attention to those of the last listed properties in which the modulus sign appears after the degree is taken. Do not forget that when placing an even power outside the logarithm sign, under the logarithm or at the base, you must leave the modulus sign.
Other beneficial properties logarithms:
The last property is very often used in complex logarithmic equations and inequalities. He should be remembered as well as everyone else, although he is often forgotten.
The simplest logarithmic equations look like:
And their solution is given by a formula that directly follows from the definition of the logarithm:
Other simple logarithmic equations are those that, using algebraic transformations and the above formulas and properties of logarithms can be reduced to the form:
The solution to such equations taking into account the ODZ is as follows:
Some others logarithmic equations with a variable at the base can be reduced to the form:
In such logarithmic equations general view the solution also follows directly from the definition of the logarithm. Only in this case there are additional restrictions for DZ that need to be taken into account. As a result, to solve a logarithmic equation with a variable in the base, you need to solve the following system:
When solving more complex logarithmic equations, which cannot be reduced to one of the equations presented above, is also actively used variable replacement method. As usual, when using this method, you need to remember that after introducing the replacement, the equation should simplify and no longer contain the old unknown. You also need to remember to perform reverse substitution of variables.
Sometimes when solving logarithmic equations you also have to use graphic method. This method is to build as accurately as possible on one coordinate plane graphs of functions that are on the left and right sides of the equation, and then find the coordinates of their intersection points from the drawing. The roots obtained in this way must be checked by substitution into the original equation.
It is also often useful when solving logarithmic equations. grouping method. When using this method, the main thing to remember is that: in order for the product of several factors to be equal to zero, it is necessary that at least one of them is equal to zero, and the rest existed. When the factors are logarithms or parentheses with logarithms, and not just parentheses with variables as in rational equations, many errors can occur. Since logarithms have many restrictions on the region where they exist.
When deciding systems of logarithmic equations most often you have to use either the substitution method or the variable replacement method. If there is such a possibility, then when solving systems of logarithmic equations, one must strive to ensure that each of the equations of the system is individually brought to a form in which it will be possible to make the transition from a logarithmic equation to a rational one.
The simplest logarithmic inequalities are solved in approximately the same way as similar equations. First, using algebraic transformations and the properties of logarithms, we must try to bring them to a form where the logarithms on the left and right sides of the inequality will have the same bases, i.e. get an inequality of the form:
After which you need to move to a rational inequality, taking into account that this transition should be performed as follows: if the base of the logarithm is greater than one, then the sign of the inequality does not need to be changed, and if the base of the logarithm is less than one, then you need to change the sign of the inequality to the opposite (this means changing "less" to "more" or vice versa). In this case, there is no need to change the minus signs to plus ones, bypassing the previously learned rules. Let's write down mathematically what we get as a result of performing such a transition. If the base is greater than one we get:
If the base of the logarithm is less than one, we change the sign of inequality and get the following system:
As we see, when solving logarithmic inequalities, as usual, the ODZ is also taken into account (this is the third condition in the systems above). Moreover, in this case it is possible not to require the positivity of both sublogarithmic expressions, but rather to require only the positivity of the smaller of them.
When deciding logarithmic inequalities with a variable at the base logarithm, it is necessary to independently consider both options (when the base is less than one and greater than one) and combine the solutions of these cases into a set. At the same time, we must not forget about DL, i.e. about the fact that both the base and all sublogarithmic expressions must be positive. Thus, when solving an inequality of the form:
We obtain the following set of systems:
More complex logarithmic inequalities can also be solved using changes of variables. Some other logarithmic inequalities (as well as logarithmic equations) require the procedure of taking the logarithm of both sides of the inequality or equation to be solved. same basis. So, when carrying out such a procedure with logarithmic inequalities, there is a subtlety. Please note that when taking logarithms to a base greater than one, the inequality sign does not change, but if the base is less than one, then the inequality sign is reversed.
If a logarithmic inequality cannot be reduced to a rational one or solved using a substitution, then in this case one must use generalized interval method, which is as follows:
- Define DL;
- Transform the inequality so that there is a zero on the right side (on the left side, if possible, reduce to common denominator, factorize, etc.);
- Find all the roots of the numerator and denominator and plot them on the number axis, and if the inequality is not strict, paint over the roots of the numerator, but in any case leave the roots of the denominator as dotted out;
- Find the sign of the entire expression on each of the intervals by substituting a number from a given interval into the transformed inequality. In this case, it is no longer possible to alternate signs in any way when passing through points on the axis. It is necessary to determine the sign of an expression on each interval by substituting the value from the interval into this expression, and so on for each interval. This is no longer possible (this is, by and large, the difference between the generalized interval method and the usual one);
- Find the intersection of the ODZ and intervals that satisfy the inequality, but do not lose individual points that satisfy the inequality (the roots of the numerator in non-strict inequalities), and do not forget to exclude from the answer all the roots of the denominator in all inequalities.
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An inequality is called logarithmic if it contains a logarithmic function.
Methods for solving logarithmic inequalities are 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 )