How to Solve Logarithmic Inequalities Lg X
There are additional topics for this topic.
Materials in Special Section 555.
For those who are strong “Not really. ”
And for those who “very much. ” )
What is a logarithmic equation?
This is an equation with logarithms. That surprised, huh?) Then I will clarify. This is an equation in which unknowns (X’s) and expressions with them are found inside the logarithms. And only there! It is important.
Here are some examples logarithmic equations:
log 2 (x1) 10 = 11 log (x1)
Note! The most diverse expressions with X are located exclusively inside the logarithms. If, suddenly, in the equation x is found somewhere outside, eg:
it will be a mixed type equation. Such equations do not have clear decision rules. We will not consider them yet. By the way, I come across equations where inside the logarithms only numbers. For instance:
What can I say? You are lucky if you come across this! Logarithm with numbers. this is some number. And that’s it. It is enough to know the properties of the logarithms to solve such an equation. Knowledge of special rules, techniques adapted specifically for solving logarithmic equations, not required here.
So, what is the logarithmic equation. figured out.
How to solve the logarithmic equations?
Decision logarithmic equations. the thing is actually not very simple. So is the section with us. on the four. A decent supply of knowledge on all sorts of related topics is required. In addition, there is a special feature in these equations. And the chip is so important that it can be safely called the main problem in solving logarithmic equations. We will deal with this problem in the next lesson in detail.
And now. do not worry. We will go the right way from simple to complex. On specific examples. The main thing is to delve into simple things and do not be lazy to follow the links, I just put them. And you will succeed. Mandatory.
Let’s start with the most elementary, simplest equations. To solve them, it is desirable to have an idea of the logarithm, but no more. Just no idea logarithm make a decision logarithmic equations. somehow awkward even. Very boldly, I would say).
The simplest logarithmic equations.
These are equations of the form:
Decision process any logarithmic equation consists in moving from an equation with logarithms to an equation without them. In the simplest equations, this transition is carried out in one step. Therefore, the simplest.)
And solving such logarithmic equations is surprisingly simple. See for yourself.
We solve the first example:
To solve this example, you don’t need to know almost anything, yes. Purely intuition!) What do we need especially don’t like in this example? I’m sorry, what. Logarithms do not like! Right. So get rid of them. We look closely at the example, and we have a natural desire. Really irresistible! Take and throw logarithms in general. And what pleases is can to do! Mathematics allows. Logarithms disappear the answer is:
Great, right? So you can (and should) do it always. Eliminating logarithms in a similar way. one of the main ways to solve logarithmic equations and inequalities. In mathematics, this operation is called potentiation. Of course, there are rules for such liquidation, but there are few of them. Remember:
It is possible to eliminate logarithms without any fear if they have:
a) the same numerical bases
c) the left-right logarithms are pure (without any coefficients) and are in splendid isolation.
I will explain the last point. In the equation, let’s say
Logarithms cannot be removed. The deuce on the right does not allow. Coefficient, you know. In the example
it is also impossible to potentiate the equation. There is no lonely logarithm on the left side. There are two of them.
In short, logarithms can be removed if the equation looks like this and only like this:
In parentheses, where the ellipsis may be any kind of expression. Simple, super complex, all sorts of. Whatever. The important thing is that after the elimination of the logarithms, we have simpler equation.It is assumed, of course, that you already know how to solve linear, square, fractional, exponential and other equations without logarithms.)
Now you can easily solve the second example:
Actually, the mind is being decided. Potentiate, we get:
Well, is it very difficult?) As you can see, logarithmic part of the solution to the equation is only in eliminating logarithms. And then comes the solution of the remaining equation without them. A trivial matter.
The following examples cannot be solved this way. Here you already need to know what the logarithm is.
We solve the third example:
We see that the logarithm is on the left:
Recall that this logarithm. some number in which you need to raise the base (i.e. seven) to get a sub-logarithm expression, i.e. (50x-1).
But this number is two! According to the equation. That is:
That, in essence, is all. Logarithm disappeared the harmless equation remains:
We solved this logarithmic equation based only on the meaning of the logarithm. What, eliminating the logarithms is still easier?) I agree. By the way, if you make a logarithm from a deuce, you can solve this example through liquidation. From any number, you can make a logarithm. over, the one we need. A very useful technique in solving logarithmic equations and (especially!) Inequalities.
Do not know how to make a logarithm !? Nothing wrong. Section 555 describes this technique in detail. You can master and apply it to the fullest! It great reduces errors.
Quite similarly (by definition), the fourth equation is solved:
To summarize this lesson. We looked at examples of the solution of the simplest logarithmic equations. It is very important. And not only because such equations are in control exams. The fact is that even the most evil and confused equations necessarily come down to the simplest!
Actually, the simplest equations. this is the finish part of the decision any equations. And this finish must be understood ironically! And further. Be sure to read this page to the end. There is a surprise there. )
We decide now on our own. We fill the hand, so to speak. )
Find the root (or the sum of the roots, if there are several) of equations:
Answers (in a mess, of course): 42; 12; 9; 25; 7; 1.5; 2; 16.
What, not everything turns out? It happens. Do not grieve! In section 555, the solution to all of these examples is detailed and clear. There you’ll certainly figure it out. Yes, and learn useful practical techniques.
Everything worked out!? All examples “one left”?) Congratulations!
It’s time to reveal the bitter truth to you. A successful solution of these examples does not guarantee success in solving all other logarithmic equations. Even the simplest ones like these. Alas.
The fact is that the solution to any logarithmic equation (even the most elementary one!) Consists of two equivalent parts. Solving the equation, and working with ODZ. One piece. solution of the equation itself. we have mastered. Not so hard right?
For this lesson, I specially selected examples in which the DLD does not affect the answer in any way. But not everyone is as kind as me, really. )
Therefore, it is necessary to master the other part as well. DLD. This is the main problem in solving logarithmic equations. And not because it is difficult. this part is even simpler than the first. But because they simply forget about ODZ. Or do not know. Or both). And fall out of the blue.
In the next lesson, we will deal with this problem. Then it will be possible to confidently decide any simple logarithmic equations and get to quite solid tasks.
If you like this site.
By the way, I have a couple more interesting sites for you.)
Here you can practice solving examples and find out your level. Testing with instant verification. We are learning. with interest!)
And here you can get acquainted with functions and derivatives.