Division of logarithms with the same bases formula. Solution of logarithmic equations. Complete Guide (2019)
The focus of this article is logarithm. Here we give the definition of the logarithm, show accepted designation, give examples of logarithms, and talk about natural and decimal logarithms. After that, consider the basic logarithmic identity.
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Definition of logarithm
The concept of a logarithm arises when solving a problem in in a certain sense inverse when you need to find the exponent known value degree and known base.
But enough preamble, it's time to answer the question "what is a logarithm"? Let us give an appropriate definition.
Definition.
Logarithm of b to base a, where a>0 , a≠1 and b>0 is the exponent to which you need to raise the number a to get b as a result.
At this stage, we note that the spoken word "logarithm" should immediately raise two ensuing questions: "what number" and "on what basis." In other words, there is simply no logarithm, but there is only the logarithm of a number in some base.
We will immediately introduce logarithm notation: the logarithm of the number b to the base a is usually denoted as log a b . The logarithm of the number b to the base e and the logarithm to the base 10 have their own special designations lnb and lgb respectively, that is, they write not log e b , but lnb , and not log 10 b , but lgb .
Now you can bring: .
And the records 
do not make sense, since in the first of them under the sign of the logarithm is a negative number, in the second - a negative number in the base, and in the third - both a negative number under the sign of the logarithm and a unit in the base.
Now let's talk about rules for reading logarithms. The entry log a b is read as "logarithm of b to base a". For example, log 2 3 is the logarithm of three to base 2, and is the logarithm of two point two thirds to base Square root out of five. The logarithm to base e is called natural logarithm, and the notation lnb is read as "the natural logarithm of b". For example, ln7 is the natural logarithm of seven, and we will read it as the natural logarithm of pi. The logarithm to base 10 also has a special name - decimal logarithm, and the notation lgb is read as "decimal logarithm b". For example, lg1 is the decimal logarithm of one, and lg2.75 is the decimal logarithm of two point seventy-five hundredths.
It is worth dwelling separately on the conditions a>0, a≠1 and b>0, under which the definition of the logarithm is given. Let us explain where these restrictions come from. To do this, we will be helped by an equality of the form, called , which directly follows from the definition of the logarithm given above.
Let's start with a≠1 . Since one is equal to one to any power, then the equality can only be true for b=1, but log 1 1 can be any real number. To avoid this ambiguity, a≠1 is accepted.
Let us substantiate the expediency of the condition a>0 . With a=0, by the definition of the logarithm, we would have equality , which is possible only with b=0 . But then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be avoided by the condition a≠0 . And for a<0 нам бы пришлось отказаться от рассмотрения рациональных и иррациональных значений логарифма, так как степень с рациональным и иррациональным показателем определена лишь для неотрицательных оснований. Поэтому и принимается условие a>0 .
Finally, the condition b>0 follows from the inequality a>0 , since , and the value of the degree with a positive base a is always positive.
In conclusion of this paragraph, we say that the voiced definition of the logarithm allows you to immediately indicate the value of the logarithm when the number under the sign of the logarithm is a certain degree of base. Indeed, the definition of the logarithm allows us to assert that if b=a p , then the logarithm of the number b to the base a is equal to p . That is, the equality log a a p =p is true. For example, we know that 2 3 =8 , then log 2 8=3 . We will talk more about this in the article.
The basic properties of the natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, expansion in a power series and representation of the function ln x by means of complex numbers are given.
Definition
natural logarithm is the function y = ln x, inverse to the exponent, x \u003d e y , and which is the logarithm to the base of the number e: ln x = log e x.
The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.
Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.
Graph of the function y = ln x.
Graph of the natural logarithm (functions y = ln x) is obtained from the graph of the exponent by mirror reflection about the straight line y = x .
The natural logarithm is defined for positive values of x . It monotonically increases on its domain of definition.
As x → 0 the limit of the natural logarithm is minus infinity ( - ∞ ).
As x → + ∞, the limit of the natural logarithm is plus infinity ( + ∞ ). For large x, the logarithm increases rather slowly. Any power function x a with a positive exponent a grows faster than the logarithm.
Properties of the natural logarithm
Domain of definition, set of values, extrema, increase, decrease
The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.
ln x values
log 1 = 0
Basic formulas for natural logarithms
Formulas arising from the definition of the inverse function:
The main property of logarithms and its consequences
Base replacement formula
Any logarithm can be expressed in terms of natural logarithms using the base change formula:
The proofs of these formulas are presented in the "Logarithm" section.
Inverse function
The reciprocal of the natural logarithm is the exponent.
If , then
If , then .
Derivative ln x
Derivative of the natural logarithm:
.
Derivative of the natural logarithm of the modulo x:
.
Derivative of the nth order:
.
Derivation of formulas > > >
Integral
The integral is calculated by integration by parts:
.
So,
Expressions in terms of complex numbers
Consider a function of a complex variable z :
.
Let's express the complex variable z via module r and argument φ
:
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If we put
, where n is an integer,
then it will be the same number for different n.
Therefore, the natural logarithm, as a function of a complex variable, is not single-valued function.
Power series expansion
For , the expansion takes place:
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
With this video, I begin a long series of lessons about logarithmic equations. Now you have three examples at once, on the basis of which we will learn to solve the simplest tasks, which are called so - protozoa.
log 0.5 (3x - 1) = -3
lg (x + 3) = 3 + 2 lg 5
Let me remind you that the simplest logarithmic equation is the following:
log a f(x) = b
It is important that the variable x is present only inside the argument, i.e. only in the function f(x). And the numbers a and b are just numbers, and in no case are functions containing the variable x.
Basic solution methods
There are many ways to solve such structures. For example, most teachers at school suggest this way: Immediately express the function f ( x ) using the formula f( x ) = a b . That is, when you meet the simplest construction, you can immediately proceed to the solution without additional actions and constructions.
Yes, of course, the decision will turn out to be correct. However, the problem with this formula is that most students do not understand, where does it come from and why exactly we raise the letter a to the letter b.
As a result, I often observe very offensive errors, when, for example, these letters are interchanged. This formula must either be understood or memorized, and the second method leads to errors at the most inopportune and most crucial moments: in exams, tests, etc.
That is why I suggest to all my students to abandon the standard school formula and use the second approach to solve logarithmic equations, which, as you probably guessed from the name, is called canonical form.
The idea of the canonical form is simple. Let's look at our task again: on the left we have log a , while the letter a means exactly the number, and in no case the function containing the variable x. Therefore, this letter is subject to all restrictions that are imposed on the base of the logarithm. namely:
1 ≠ a > 0
On the other hand, from the same equation, we see that the logarithm must be is equal to the number b , and no restrictions are imposed on this letter, because it can take any value - both positive and negative. It all depends on what values the function f(x) takes.
And here we remember our wonderful rule that any number b can be represented as a logarithm in base a from a to the power of b:
b = log a a b
How to remember this formula? Yes, very simple. Let's write the following construction:
b = b 1 = b log a a
Of course, in this case, all the restrictions that we wrote down at the beginning arise. And now let's use the basic property of the logarithm, and enter the factor b as the power of a. We get:
b = b 1 = b log a a = log a a b
As a result, the original equation will be rewritten in the following form:
log a f (x) = log a a b → f (x) = a b
That's all. The new function no longer contains a logarithm and is solved by standard algebraic techniques.
Of course, someone will now object: why was it necessary to come up with some kind of canonical formula at all, why perform two additional unnecessary steps, if it was possible to immediately go from the original construction to the final formula? Yes, if only because most students do not understand where this formula comes from and, as a result, regularly make mistakes when applying it.
But such a sequence of actions, consisting of three steps, allows you to solve the original logarithmic equation, even if you do not understand where that final formula comes from. By the way, this entry is called the canonical formula:
log a f(x) = log a a b
The convenience of the canonical form also lies in the fact that it can be used to solve a very wide class of logarithmic equations, and not just the simplest ones that we are considering today.
Solution examples
And now let's consider real examples. So let's decide:
log 0.5 (3x - 1) = -3
Let's rewrite it like this:
log 0.5 (3x − 1) = log 0.5 0.5 −3
Many students are in a hurry and try to immediately raise the number 0.5 to the power that came to us from the original problem. And indeed, when you are already well trained in solving such problems, you can immediately perform this step.
However, if now you are just starting to study this topic, it is better not to rush anywhere so as not to make offensive mistakes. So we have the canonical form. We have:
3x - 1 = 0.5 -3
This is no longer a logarithmic equation, but a linear one with respect to the variable x. To solve it, let's first deal with the number 0.5 to the power of −3. Note that 0.5 is 1/2.
(1/2) −3 = (2/1) 3 = 8
All decimals convert to normal when you solve a logarithmic equation.
We rewrite and get:
3x − 1 = 8
3x=9
x=3
All we got the answer. The first task is solved.
Second task
Let's move on to the second task:
As you can see, this equation is no longer the simplest one. If only because the difference is on the left, and not a single logarithm in one base.
Therefore, you need to somehow get rid of this difference. In this case, everything is very simple. Let's take a closer look at the bases: on the left is the number under the root:
General recommendation: in all logarithmic equations, try to get rid of radicals, i.e., entries with roots, and move on to power functions, simply because the exponents of these powers are easily taken out of the sign of the logarithm, and in the end, such a notation greatly simplifies and speeds up calculations. Let's write it like this:
Now we recall the remarkable property of the logarithm: from the argument, as well as from the base, you can take out degrees. In the case of bases, the following happens:
log a k b = 1/k loga b
In other words, the number that stood in the degree of the base is brought forward and at the same time turned over, that is, it becomes the reciprocal of the number. In our case, there was a degree of base with an indicator of 1/2. Therefore, we can take it out as 2/1. We get:
5 2 log 5 x − log 5 x = 18
10 log 5 x − log 5 x = 18
Please note: in no case should you get rid of logarithms at this step. Think back to grade 4-5 math and the order of operations: multiplication is performed first, and only then addition and subtraction are performed. In this case, we subtract one of the same elements from 10 elements:
9 log 5 x = 18
log 5 x = 2
Now our equation looks like it should. This is simplest design, and we solve it with the canonical form:
log 5 x = log 5 5 2
x = 5 2
x=25
That's all. The second problem is solved.
Third example
Let's move on to the third task:
lg (x + 3) = 3 + 2 lg 5
Recall the following formula:
log b = log 10 b
If for some reason you are confused by writing lg b , then when doing all the calculations, you can simply write log 10 b . You can work with decimal logarithms in the same way as with others: take out powers, add, and represent any number as lg 10.
It is precisely these properties that we will now use to solve the problem, since it is not the simplest one that we wrote down at the very beginning of our lesson.
To begin with, note that the factor 2 before lg 5 can be inserted and becomes a power of base 5. In addition, the free term 3 can also be represented as a logarithm - this is very easy to observe from our notation.
Judge for yourself: any number can be represented as log to base 10:
3 = log 10 10 3 = log 10 3
Let's rewrite the original problem taking into account the received changes:
lg (x − 3) = lg 1000 + lg 25
lg (x − 3) = lg 1000 25
lg (x - 3) = lg 25 000
Before us is again the canonical form, and we obtained it bypassing the stage of transformations, i.e., the simplest logarithmic equation did not come up anywhere with us.
That's what I was talking about at the very beginning of the lesson. The canonical form allows solving a wider class of problems than the standard one. school formula given by most school teachers.
That's all, we get rid of the sign of the decimal logarithm, and we get a simple linear construction:
x + 3 = 25,000
x = 24997
All! Problem solved.
A note about scope
Here I would like to make an important remark about the domain of definition. Surely now there are students and teachers who will say: “When we solve expressions with logarithms, it is imperative to remember that the argument f (x) must be greater than zero!” In this regard, a logical question arises: why in none of the considered problems did we require that this inequality be satisfied?
Do not worry. No extra roots will appear in these cases. And this is another great trick that allows you to speed up the solution. Just know that if in the problem the variable x occurs only in one place (or rather, in the one and only argument of the one and only logarithm), and nowhere else in our case does the variable x, then write the domain not necessary because it will run automatically.
Judge for yourself: in the first equation, we got that 3x - 1, i.e., the argument should be equal to 8. This automatically means that 3x - 1 will be greater than zero.
With the same success, we can write that in the second case, x must be equal to 5 2, i.e., it is certainly greater than zero. And in the third case, where x + 3 = 25,000, i.e., again, obviously greater than zero. In other words, the scope is automatic, but only if x occurs only in the argument of only one logarithm.
That's all you need to know to solve simple problems. This rule alone, together with the transformation rules, will allow you to solve a very wide class of problems.
But let's be honest: in order to finally deal with this technique, in order to learn how to apply the canonical form logarithmic equation It's not enough just to watch one video tutorial. Therefore, right now, download the options for an independent solution that are attached to this video tutorial and start solving at least one of these two independent works.
It will take you just a few minutes. But the effect of such training will be much higher compared to if you just watched this video tutorial.
I hope this lesson will help you understand logarithmic equations. Apply the canonical form, simplify expressions using the rules for working with logarithms - and you will not be afraid of any tasks. And that's all I have for today.
Scope consideration
Now let's talk about the domain of the logarithmic function, as well as how this affects the solution of logarithmic equations. Consider a construction of the form
log a f(x) = b
Such an expression is called the simplest - it has only one function, and the numbers a and b are just numbers, and in no case are a function that depends on the variable x. It is solved very simply. You just need to use the formula:
b = log a a b
This formula is one of the key properties of the logarithm, and when substituting into our original expression, we get the following:
log a f(x) = log a a b
f(x) = a b
This is already a familiar formula from school textbooks. Many students will probably have a question: since the function f ( x ) in the original expression is under the log sign, the following restrictions are imposed on it:
f(x) > 0
This restriction is valid because the logarithm of negative numbers does not exist. So, maybe because of this limitation, you should introduce a check for answers? Perhaps they need to be substituted in the source?
No, in the simplest logarithmic equations, an additional check is unnecessary. And that's why. Take a look at our final formula:
f(x) = a b
The fact is that the number a in any case is greater than 0 - this requirement is also imposed by the logarithm. The number a is the base. In this case, no restrictions are imposed on the number b. But this does not matter, because no matter what degree we raise a positive number, we will still get a positive number at the output. Thus, the requirement f (x) > 0 is fulfilled automatically.
What is really worth checking is the scope of the function under the log sign. There can be quite complex designs, and in the process of solving them, you must definitely follow them. Let's get a look.
First task:
First step: convert the fraction on the right. We get:
We get rid of the sign of the logarithm and get the usual irrational equation:
Of the obtained roots, only the first one suits us, since the second root is less than zero. The only answer will be the number 9. That's it, the problem is solved. No additional checks that the expression under the logarithm sign is greater than 0 are required, because it is not just greater than 0, but by the condition of the equation it is equal to 2. Therefore, the requirement "greater than zero" is automatically fulfilled.
Let's move on to the second task:
Everything is the same here. We rewrite the construction, replacing the triple:
We get rid of the signs of the logarithm and get an irrational equation:
We square both parts, taking into account the restrictions, and we get:
4 - 6x - x 2 = (x - 4) 2
4 - 6x - x 2 = x 2 + 8x + 16
x2 + 8x + 16 −4 + 6x + x2 = 0
2x2 + 14x + 12 = 0 |:2
x2 + 7x + 6 = 0
We solve the resulting equation through the discriminant:
D \u003d 49 - 24 \u003d 25
x 1 = -1
x 2 \u003d -6
But x = −6 does not suit us, because if we substitute this number into our inequality, we get:
−6 + 4 = −2 < 0
In our case, it is required that it be greater than 0 or, in extreme cases, equal. But x = −1 suits us:
−1 + 4 = 3 > 0
The only answer in our case is x = −1. That's all the solution. Let's go back to the very beginning of our calculations.
The main conclusion from this lesson is that it is not required to check the limits for a function in the simplest logarithmic equations. Because in the process of solving all the constraints are executed automatically.
However, this by no means means that you can forget about verification altogether. In the process of working on a logarithmic equation, it may well turn into an irrational one, which will have its own limitations and requirements for the right side, which we have seen today in two different examples.
Feel free to solve such problems and be especially careful if there is a root in the argument.
Logarithmic equations with different bases
We continue to study logarithmic equations and analyze two more rather interesting tricks with which it is fashionable to solve more complex structures. But first, let's remember how the simplest tasks are solved:
log a f(x) = b
In this notation, a and b are just numbers, and in the function f (x) the variable x must be present, and only there, that is, x must be only in the argument. We will transform such logarithmic equations using the canonical form. For this, we note that
b = log a a b
And a b is just an argument. Let's rewrite this expression as follows:
log a f(x) = log a a b
This is exactly what we are trying to achieve, so that both on the left and on the right there is a logarithm to the base a. In this case, we can, figuratively speaking, cross out the signs of log, and from the point of view of mathematics, we can say that we simply equate the arguments:
f(x) = a b
As a result, we get a new expression that will be solved much easier. Let's apply this rule to our tasks today.
So the first design:
First of all, I note that there is a fraction on the right, the denominator of which is log. When you see an expression like this, it's worth remembering the wonderful property of logarithms:
Translated into Russian, this means that any logarithm can be represented as a quotient of two logarithms with any base c. Of course, 0< с ≠ 1.
So: this formula has one wonderful special case when the variable c is equal to the variable b. In this case, we get a construction of the form:
It is this construction that we observe from the sign on the right in our equation. Let's replace this construction with log a b , we get:
In other words, in comparison with the original task, we have swapped the argument and the base of the logarithm. Instead, we had to flip the fraction.
We recall that any degree can be taken out of the base according to the following rule:
In other words, the coefficient k, which is the degree of the base, is taken out as an inverted fraction. Let's take it out as an inverted fraction:
The fractional factor cannot be left in front, because in this case we will not be able to represent this entry as a canonical form (after all, in the canonical form, there is no additional factor in front of the second logarithm). Therefore, let's put the fraction 1/4 in the argument as a power:
Now we equate the arguments whose bases are the same (and we really have the same bases), and write:
x + 5 = 1
x = −4
That's all. We got the answer to the first logarithmic equation. Pay attention: in the original problem, the variable x occurs only in one log, and it is in its argument. Therefore, there is no need to check the domain, and our number x = −4 is indeed the answer.
Now let's move on to the second expression:
log 56 = log 2 log 2 7 − 3 log (x + 4)
Here, in addition to the usual logarithms, we will have to work with lg f (x). How to solve such an equation? It may seem to an unprepared student that this is some kind of tin, but in fact everything is solved elementarily.
Look closely at the term lg 2 log 2 7. What can we say about it? The bases and arguments of log and lg are the same, and this should give some clues. Let's remember once again how the degrees are taken out from under the sign of the logarithm:
log a b n = n log a b
In other words, what was the power of the number b in the argument becomes a factor in front of log itself. Let's apply this formula to the expression lg 2 log 2 7. Don't be afraid of lg 2 - this is the most common expression. You can rewrite it like this:
For him, all the rules that apply to any other logarithm are valid. In particular, the factor in front can be introduced into the power of the argument. Let's write:
Very often, students point blank do not see this action, because it is not good to enter one log under the sign of another. In fact, there is nothing criminal in this. Moreover, we get a formula that is easy to calculate if you remember an important rule:
This formula can be considered both as a definition and as one of its properties. In any case, if you convert a logarithmic equation, you should know this formula in the same way as the representation of any number in the form of log.
We return to our task. We rewrite it taking into account the fact that the first term to the right of the equal sign will simply be equal to lg 7. We have:
lg 56 = lg 7 − 3lg (x + 4)
Let's move lg 7 to the left, we get:
lg 56 - lg 7 = -3lg (x + 4)
We subtract the expressions on the left because they have the same base:
lg (56/7) = -3lg (x + 4)
Now let's take a closer look at the equation we've got. It is practically the canonical form, but there is a factor −3 on the right. Let's put it in the right lg argument:
lg 8 = lg (x + 4) −3
Before us is the canonical form of the logarithmic equation, so we cross out the signs of lg and equate the arguments:
(x + 4) -3 = 8
x + 4 = 0.5
That's all! We have solved the second logarithmic equation. In this case, no additional checks are required, because in the original problem x was present in only one argument.
Let me recap the key points of this lesson.
The main formula that is studied in all the lessons on this page devoted to solving logarithmic equations is the canonical form. And don't be put off by the fact that most school textbooks teach you how to solve these kinds of problems differently. This tool works very efficiently and allows you to solve a much wider class of problems than the simplest ones that we studied at the very beginning of our lesson.
In addition, to solve logarithmic equations, it will be useful to know the basic properties. Namely:
- The formula for moving to one base and a special case when we flip log (this was very useful to us in the first task);
- The formula for bringing in and taking out powers from under the sign of the logarithm. Here, many students get stuck and do not see point-blank that the power taken out and brought in can itself contain log f (x). Nothing wrong with that. We can introduce one log according to the sign of another and at the same time significantly simplify the solution of the problem, which is what we observe in the second case.
In conclusion, I would like to add that it is not required to check the scope in each of these cases, because everywhere the variable x is present in only one sign of log, and at the same time is in its argument. As a consequence, all domain requirements are met automatically.
Problems with variable base
Today we will consider logarithmic equations, which for many students seem non-standard, if not completely unsolvable. It's about about expressions based not on numbers, but on variables and even functions. We will solve such constructions using our standard technique, namely, through the canonical form.
To begin with, let's recall how the simplest problems are solved, which are based on ordinary numbers. So, the simplest construction is called
log a f(x) = b
To solve such problems, we can use the following formula:
b = log a a b
We rewrite our original expression and get:
log a f(x) = log a a b
Then we equate the arguments, i.e. we write:
f(x) = a b
Thus, we get rid of the log sign and solve the usual problem. In this case, the roots obtained in the solution will be the roots of the original logarithmic equation. In addition, the record, when both the left and the right are on the same logarithm with the same base, is called the canonical form. It is to this record that we will try to reduce today's constructions. So let's go.
First task:
log x − 2 (2x 2 − 13x + 18) = 1
Replace 1 with log x − 2 (x − 2) 1 . The degree that we observe in the argument is, in fact, the number b , which was to the right of the equal sign. So let's rewrite our expression. We get:
log x - 2 (2x 2 - 13x + 18) = log x - 2 (x - 2)
What do we see? Before us is the canonical form of the logarithmic equation, so we can safely equate the arguments. We get:
2x2 - 13x + 18 = x - 2
But the solution does not end there, because given equation not equivalent to the original. After all, the resulting construction consists of functions that are defined on the entire number line, and our original logarithms are not defined everywhere and not always.
Therefore, we must write down the domain of definition separately. Let's not be wiser and first write down all the requirements:
First, the argument of each of the logarithms must be greater than 0:
2x 2 − 13x + 18 > 0
x − 2 > 0
Secondly, the base must not only be greater than 0, but also different from 1:
x − 2 ≠ 1
As a result, we get the system:
But don't be alarmed: when processing logarithmic equations, such a system can be greatly simplified.
Judge for yourself: on the one hand, we are required that the quadratic function be greater than zero, and on the other hand, this quadratic function is equated to a certain linear expression, which is also required that it be greater than zero.
In this case, if we require that x − 2 > 0, then the requirement 2x 2 − 13x + 18 > 0 will also be automatically satisfied. Therefore, we can safely cross out the inequality containing a quadratic function. Thus, the number of expressions contained in our system will be reduced to three.
Of course, we might as well cross out linear inequality, i.e. cross out x − 2 > 0 and require that 2x 2 − 13x + 18 > 0. But you must agree that it is much faster and easier to solve the simplest linear inequality than this system we get the same roots.
In general, try to optimize calculations whenever possible. And in the case of logarithmic equations, cross out the most difficult inequalities.
Let's rewrite our system:
Here is such a system of three expressions, two of which we, in fact, have already figured out. Let's separately write out the quadratic equation and solve it:
2x2 - 14x + 20 = 0
x2 − 7x + 10 = 0
Before us is a reduced square trinomial and, therefore, we can use the Vieta formulas. We get:
(x − 5)(x − 2) = 0
x 1 = 5
x2 = 2
Now, back to our system, we find that x = 2 doesn't suit us, because we're required to have x strictly greater than 2.
But x \u003d 5 suits us quite well: the number 5 is greater than 2, and at the same time 5 is not equal to 3. Therefore, the only solution of this system will be x = 5.
Everything, the task is solved, including taking into account the ODZ. Let's move on to the second equation. Here we are waiting for more interesting and meaningful calculations:
The first step: as well as last time, we bring all this business to a canonical form. To do this, we can write the number 9 as follows:
The base with the root can not be touched, but it is better to transform the argument. Let's move from the root to the power with a rational exponent. Let's write:
Let me not rewrite our whole big logarithmic equation, but just immediately equate the arguments:
x 3 + 10x 2 + 31x + 30 = x 3 + 9x 2 + 27x + 27
x 2 + 4x + 3 = 0
Before us is the again reduced square trinomial, we will use the Vieta formulas and write:
(x + 3)(x + 1) = 0
x 1 = -3
x 2 = -1
So, we got the roots, but no one guaranteed us that they would fit the original logarithmic equation. After all, log signs impose additional restrictions (here we would have to write down the system, but due to the cumbersomeness of the whole construction, I decided to calculate the domain of definition separately).
First of all, remember that the arguments must be greater than 0, namely:
These are the requirements imposed by the domain of definition.
We note right away that since we equate the first two expressions of the system to each other, we can cross out any of them. Let's cross out the first one because it looks more menacing than the second one.
In addition, note that the solutions of the second and third inequalities will be the same sets (the cube of some number is greater than zero, if this number itself is greater than zero; similarly with the root of the third degree - these inequalities are completely similar, so one of them we can cross it out).
But with the third inequality, this will not work. Let's get rid of the sign of the radical on the left, for which we raise both parts to a cube. We get:
So we get the following requirements:
−2 ≠ x > −3
Which of our roots: x 1 = -3 or x 2 = -1 meets these requirements? Obviously, only x = −1, because x = −3 does not satisfy the first inequality (because our inequality is strict). In total, returning to our problem, we get one root: x = −1. That's it, problem solved.
Once again, the key points of this task:
- Feel free to apply and solve logarithmic equations using canonical form. Students who make such a record, and do not go directly from the original problem to a construction like log a f ( x ) = b , make much fewer errors than those who are in a hurry somewhere, skipping intermediate steps of calculations;
- As soon as a variable base appears in the logarithm, the problem ceases to be the simplest. Therefore, when solving it, it is necessary to take into account the domain of definition: the arguments must be greater than zero, and the bases must not only be greater than 0, but they must also not be equal to 1.
You can impose the last requirements on the final answers in different ways. For example, it is possible to solve a whole system containing all domain requirements. On the other hand, you can first solve the problem itself, and then remember about the domain of definition, work it out separately in the form of a system and apply it to the obtained roots.
Which way to choose when solving a particular logarithmic equation is up to you. In any case, the answer will be the same.
With the development of society, the complexity of production, mathematics also developed. Movement from simple to complex. From the usual accounting method of addition and subtraction, with their repeated repetition, they came to the concept of multiplication and division. The reduction of the multiply repeated operation became the concept of exponentiation. The first tables of the dependence of numbers on the base and the number of exponentiation were compiled back in the 8th century by the Indian mathematician Varasena. From them, you can count the time of occurrence of logarithms.
Historical outline
The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation associated with multiplication and division of multi-digit numbers. The ancient tables did a great service. They allowed to replace complex operations to simpler ones - addition and subtraction. A big step forward was the work of the mathematician Michael Stiefel, published in 1544, in which he realized the idea of many mathematicians. This made it possible to use tables not only for degrees in the form of prime numbers, but also for arbitrary rational ones.
In 1614, the Scotsman John Napier, developing these ideas, first introduced the new term "logarithm of a number." New complex tables for calculating the logarithms of sines and cosines, as well as tangents. This greatly reduced the work of astronomers.
New tables began to appear, which were successfully used by scientists for three centuries. A lot of time passed before the new operation in algebra acquired its finished form. The logarithm was defined and its properties were studied.
Only in the 20th century, with the advent of the calculator and the computer, mankind abandoned the ancient tables that had been successfully operating throughout the 13th centuries.
Today we call the logarithm of b to base a the number x, which is the power of a, to get the number b. This is written as a formula: x = log a(b).
For example, log 3(9) will be equal to 2. This is obvious if you follow the definition. If we raise 3 to the power of 2, we get 9.
Thus, the formulated definition puts only one restriction, the numbers a and b must be real.
Varieties of logarithms
The classical definition is called the real logarithm and is actually a solution to the equation a x = b. The option a = 1 is borderline and is of no interest. Note: 1 to any power is 1.
Real value of the logarithm defined only if the base and the argument is greater than 0, and the base must not be equal to 1.
Special place in the field of mathematics play logarithms, which will be named depending on the value of their base:
Rules and restrictions
The fundamental property of logarithms is the rule: the logarithm of a product is equal to the logarithmic sum. log abp = log a(b) + log a(p).
As a variant of this statement, it will be: log c (b / p) \u003d log c (b) - log c (p), the quotient function is equal to the difference of the functions.
It is easy to see from the previous two rules that: log a(b p) = p * log a(b).
Other properties include:
Comment. Do not make a common mistake - the logarithm of the sum is not is equal to the sum logarithms.
For many centuries, the operation of finding the logarithm was a rather time-consuming task. Mathematicians used the well-known formula of the logarithmic theory of expansion into a polynomial:
ln (1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... + ((-1)^(n + 1))*(( x^n)/n), where n is a natural number greater than 1, which determines the accuracy of the calculation.
Logarithms with other bases were calculated using the theorem on the transition from one base to another and the property of the logarithm of the product.
Since this method is very laborious and when solving practical problems difficult to implement, they used pre-compiled tables of logarithms, which greatly accelerated the entire work.
In some cases, specially compiled graphs of logarithms were used, which gave less accuracy, but significantly speeded up the search. desired value. The curve of the function y = log a(x), built on several points, allows using the usual ruler to find the values of the function at any other point. Engineers long time for these purposes, the so-called graph paper was used.
In the 17th century, the first auxiliary analog computing conditions appeared, which to XIX century acquired a finished look. The most successful device was called the slide rule. Despite the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and this is difficult to overestimate. Currently, few people are familiar with this device.
The advent of calculators and computers made it pointless to use any other devices.
Equations and inequalities
The following formulas are used to solve various equations and inequalities using logarithms:
- Transition from one base to another: log a(b) = log c(b) / log c(a);
- As a consequence of the previous version: log a(b) = 1 / log b(a).
To solve inequalities, it is useful to know:
- The value of the logarithm will be positive only if the base and the argument are both greater than or less than one; if at least one condition is violated, the value of the logarithm will be negative.
- If the logarithm function is applied to the right and left sides of the inequality, and the base of the logarithm is greater than one, then the sign of the inequality is preserved; otherwise, it changes.
Task examples
Consider several options for using logarithms and their properties. Examples with solving equations:
Consider the option of placing the logarithm in the degree:
- Task 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the notation is similar to the following (5^2)^log5(3) or 5^(2 * log 5(3)). Let's write it differently: 5^log 5(3*2), or the square of a number as a function argument can be written as the square of the function itself (5^log 5(3))^2. Using the properties of logarithms, this expression is 3^2. Answer: as a result of the calculation we get 9.
Practical use
Being a purely mathematical tool, it seems far from real life that the logarithm suddenly acquired great importance to describe objects real world. It is difficult to find a science where it is not used. This fully applies not only to natural, but also humanitarian areas knowledge.
Logarithmic dependencies
Here are some examples of numerical dependencies:
Mechanics and physics
Historically, mechanics and physics have always developed using mathematical methods research and at the same time served as an incentive for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. We give only two examples of the description of physical laws using the logarithm.
It is possible to solve the problem of calculating such a complex quantity as the speed of a rocket using the Tsiolkovsky formula, which laid the foundation for the theory of space exploration:
V = I * ln(M1/M2), where
- V is the final speed of the aircraft.
- I is the specific impulse of the engine.
- M 1 is the initial mass of the rocket.
- M 2 - final mass.
Another important example - this is the use in the formula of another great scientist, Max Planck, which serves to evaluate the equilibrium state in thermodynamics.
S = k * ln (Ω), where
- S is a thermodynamic property.
- k is the Boltzmann constant.
- Ω is the statistical weight of different states.
Chemistry
Less obvious would be the use of formulas in chemistry containing the ratio of logarithms. Here are just two examples:
- The Nernst equation, the condition of the redox potential of the medium in relation to the activity of substances and the equilibrium constant.
- The calculation of such constants as the autoprolysis index and the acidity of the solution is also not complete without our function.
Psychology and biology
And it’s completely incomprehensible what the psychology has to do with it. It turns out that the strength of sensation is well described by this function as the inverse ratio of the intensity of the stimulus to lower value intensity.
After the above examples, it is no longer surprising that the theme of logarithms is also widely used in biology. Pro biological forms, corresponding to logarithmic spirals, you can write entire volumes.
Other areas
It seems that the existence of the world is impossible without connection with this function, and it governs all laws. Especially when the laws of nature are connected with a geometric progression. It is worth referring to the MatProfi website, and there are many such examples in the following areas of activity:
The list could be endless. Having mastered the basic laws of this function, you can plunge into the world of infinite wisdom.
(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") numbers b by reason a(log α b) is called such a number c, and b= a c, that is, log α b=c and b=ac are equivalent. The logarithm makes sense if a > 0, a ≠ 1, b > 0.
In other words logarithm numbers b by reason a formulated as an exponent to which a number must be raised a to get the number b(the logarithm exists only for positive numbers).
From this formulation it follows that the calculation x= log α b, is equivalent to solving the equation a x =b.
For example:
log 2 8 = 3 because 8=2 3 .
We note that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value when the number under the sign of the logarithm is a certain power of the base. Indeed, the formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b by reason a equals with. It is also clear that the topic of logarithm is closely related to the topic degree of number.
The calculation of the logarithm is referred to logarithm. The logarithm is mathematical operation taking the logarithm. When taking a logarithm, the products of factors are transformed into sums of terms.
Potentiation is the mathematical operation inverse to logarithm. When potentiating, the given base is raised to the power of the expression on which the potentiation is performed. In this case, the sums of terms are transformed into the product of factors.
Quite often, real logarithms with bases 2 (binary), e Euler number e ≈ 2.718 (natural logarithm) and 10 (decimal) are used.
At this stage, it is worth considering samples of logarithms log 7 2 , ln √ 5, lg0.0001.
And the entries lg (-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the sign of the logarithm, in the second - a negative number in the base, and in the third - and a negative number under the sign of the logarithm and unit in the base.
Conditions for determining the logarithm.
It is worth considering separately the conditions a > 0, a ≠ 1, b > 0. definition of a logarithm. Let's consider why these restrictions are taken. This will help us with an equality of the form x = log α b, called the basic logarithmic identity, which directly follows from the definition of the logarithm given above.
Take the condition a≠1. Since one is equal to one to any power, then the equality x=log α b can only exist when b=1, but log 1 1 will be any real number. To eliminate this ambiguity, we take a≠1.
Let us prove the necessity of the condition a>0. At a=0 according to the formulation of the logarithm, can only exist when b=0. And then accordingly log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. To eliminate this ambiguity, the condition a≠0. And when a<0 we would have to reject the analysis of rational and irrational values of the logarithm, since the exponent with rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition a>0.
And the last condition b>0 follows from the inequality a>0, since x=log α b, and the value of the degree with a positive base a always positive.
Features of logarithms.
Logarithms characterized by distinctive features, which led to their widespread use to greatly facilitate painstaking calculations. In the transition "to the world of logarithms", multiplication is transformed into much easier addition, division into subtraction, and exponentiation and root extraction are transformed into multiplication and division by the exponent, respectively.
The formulation of logarithms and a table of their values (for trigonometric functions) was first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, enlarged and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until electronic calculators and computers began to be used.