In relation to
the task of finding any of the three numbers from the other two given ones can be set. If a and then N are given, they are found by exponentiation. If N and then a are given by taking the root of the degree x (or raising it to the power). Now consider the case when, given a and N, we need to find x.
Let the number N be positive: the number a be positive and not equal to one: .
Definition. The logarithm of the number N to the base a is the exponent to which a must be raised to obtain the number N; logarithm is denoted by
Thus, in equality (26.1) the exponent is found as the logarithm of N to base a. Posts
have same meaning. Equality (26.1) is sometimes called the main identity of the theory of logarithms; in reality it expresses the definition of the concept of logarithm. By this definition The base of the logarithm a is always positive and different from unity; the logarithmic number N is positive. Negative numbers and zero have no logarithms. It can be proven that any number with a given base has a well-defined logarithm. Therefore equality entails . Note that the essential condition here is otherwise the conclusion would not be justified, since the equality is true for any values of x and y.
Example 1. Find
Solution. To obtain a number, you must raise the base 2 to the power Therefore.
You can make notes when solving such examples in the following form:
Example 2. Find .
Solution. We have
In examples 1 and 2, we easily found the desired logarithm by representing the logarithm number as a power of the base with rational indicator. IN general case, for example, for etc., this cannot be done, since the logarithm has irrational meaning. Let us pay attention to one issue related to this statement. In paragraph 12 we gave the concept of the possibility of determining any real degree given positive number. This was necessary for the introduction of logarithms, which, generally speaking, can be irrational numbers.
Let's look at some properties of logarithms.
Property 1. If the number and base are equal, then the logarithm equal to one, and, conversely, if the logarithm is equal to one, then the number and base are equal.
Proof. Let By the definition of a logarithm we have and whence
Conversely, let Then by definition
Property 2. The logarithm of one to any base is equal to zero.
Proof. By definition of logarithm ( zero degree any positive base is equal to one, see (10.1)). From here
Q.E.D.
The converse statement is also true: if , then N = 1. Indeed, we have .
Before formulating the next property of logarithms, let us agree to say that two numbers a and b lie on the same side of the third number c if they are both greater than c or less than c. If one of these numbers is greater than c, and the other is less than c, then we will say that they lie along different sides from the village
Property 3. If the number and base lie on the same side of one, then the logarithm is positive; If the number and base lie on opposite sides of one, then the logarithm is negative.
The proof of property 3 is based on the fact that the power of a is greater than one if the base is greater than one and the exponent is positive or the base is less than one and the exponent is negative. A power is less than one if the base is greater than one and the exponent is negative or the base is less than one and the exponent is positive.
There are four cases to consider:
We will limit ourselves to analyzing the first of them; the reader will consider the rest on his own.
Let then in equality the exponent can be neither negative nor equal to zero, therefore, it is positive, i.e., as required to be proved.
Example 3. Find out which of the logarithms below are positive and which are negative:
Solution, a) since the number 15 and the base 12 are located on the same side of one;
b) since 1000 and 2 are located on one side of the unit; in this case, it is not important that the base is greater than the logarithmic number;
c) since 3.1 and 0.8 lie on opposite sides of unity;
G) ; Why?
d) ; Why?
The following properties 4-6 are often called the rules of logarithmation: they allow, knowing the logarithms of some numbers, to find the logarithms of their product, quotient, and degree of each of them.
Property 4 (product logarithm rule). Logarithm of the product of several positive numbers by this basis equal to the sum logarithms of these numbers to the same base.
Proof. Let the given numbers be positive.
For the logarithm of their product, we write the equality (26.1) that defines the logarithm:
From here we will find
Comparing the exponents of the first and last expressions, we obtain the required equality:
Note that the condition is essential; logarithm of the product of two negative numbers makes sense, but in this case we get
In general, if the product of several factors is positive, then its logarithm is equal to the sum of the logarithms of the absolute values of these factors.
Property 5 (rule for taking logarithms of quotients). The logarithm of a quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor, taken to the same base. Proof. We consistently find
Q.E.D.
Property 6 (power logarithm rule). Logarithm of the power of some positive number equal to the logarithm this number multiplied by the exponent.
Proof. Let us write again the main identity (26.1) for the number:
Q.E.D.
Consequence. The logarithm of a root of a positive number is equal to the logarithm of the radical divided by the exponent of the root:
The validity of this corollary can be proven by imagining how and using property 6.
Example 4. Take logarithm to base a:
a) (it is assumed that all values b, c, d, e are positive);
b) (it is assumed that ).
Solution, a) It is convenient to go to fractional powers in this expression:
Based on equalities (26.5)-(26.7), we can now write:
We notice that simpler operations are performed on the logarithms of numbers than on the numbers themselves: when multiplying numbers, their logarithms are added, when dividing, they are subtracted, etc.
That is why logarithms are used in computing practice (see paragraph 29).
The inverse action of logarithm is called potentiation, namely: potentiation is the action by which the number itself is found from a given logarithm of a number. Essentially, potentiation is not any special action: it comes down to raising a base to a power ( equal to the logarithm numbers). The term "potentiation" can be considered synonymous with the term "exponentiation".
When potentiating, you must use the rules inverse to the rules of logarithmation: replace the sum of logarithms with the logarithm of the product, the difference of logarithms with the logarithm of the quotient, etc. In particular, if there is a factor in front of the sign of the logarithm, then during potentiation it must be transferred to the exponent degrees under the sign of the logarithm.
Example 5. Find N if it is known that
Solution. In connection with the just stated rule of potentiation, we will transfer the factors 2/3 and 1/3 standing in front of the signs of logarithms on the right side of this equality into exponents under the signs of these logarithms; we get
Now we replace the difference of logarithms with the logarithm of the quotient:
to obtain the last fraction in this chain of equalities, we freed the previous fraction from irrationality in the denominator (clause 25).
Property 7. If the base is greater than one, then larger number has a larger logarithm (and a smaller number has a smaller one), if the base is less than one, then a larger number has a smaller logarithm (and a smaller number has a larger one).
This property is also formulated as a rule for taking logarithms of inequalities, both sides of which are positive:
When taking logarithms of inequalities to the base, greater than one, the sign of inequality is preserved, and when taking a logarithm to a base less than one, the sign of inequality changes to the opposite (see also paragraph 80).
The proof is based on properties 5 and 3. Consider the case when If , then and, taking logarithms, we obtain
(a and N/M lie on the same side of unity). From here
Case a follows, the reader will figure it out on his own.
Follows from its definition. And so the logarithm of the number b based on A is defined as the exponent to which a number must be raised a to get the number b(logarithm exists only for positive numbers).
From this formulation it follows that the calculation x=log a b, is equivalent to solving the equation a x =b. For example, log 2 8 = 3 because 8 = 2 3 . The formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b based on a equals With. It is also clear that the topic of logarithms is closely related to the topic powers of a number.
With logarithms, as with any numbers, you can do addition operations, subtraction and transform in every possible way. But due to the fact that logarithms are not entirely ordinary numbers, their own special rules apply here, which are called main properties.
Adding and subtracting logarithms.
Let's take two logarithms with on the same grounds: log a x And log a y. Then it is possible to perform addition and subtraction operations:
log a x+ log a y= log a (x·y);
log a x - log a y = log a (x:y).
log a(x 1 . x 2 . x 3 ... x k) = log a x 1 + log a x 2 + log a x 3 + ... + log a x k.
From logarithm quotient theorem One more property of the logarithm can be obtained. It is common knowledge that log a 1= 0, therefore
log a 1 /b=log a 1 - log a b= - log a b.
This means there is an equality:
log a 1 / b = - log a b.
Logarithms of two reciprocal numbers for the same reason will differ from each other solely by sign. So:
Log 3 9= - log 3 1 / 9 ; log 5 1 / 125 = -log 5 125.
What is a logarithm?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially equations with logarithms.
This is absolutely not true. Absolutely! Don't believe me? Fine. Now, in just 10 - 20 minutes you:
1. You will understand what is a logarithm.
2. Learn to solve a whole class exponential equations. Even if you haven't heard anything about them.
3. Learn to calculate simple logarithms.
Moreover, for this you will only need to know the multiplication table and how to raise a number to a power...
I feel like you have doubts... Well, okay, mark the time! Go!
First, solve this equation in your head:
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Instructions
Write down the given logarithmic expression. If the expression uses the logarithm of 10, then its notation is shortened and looks like this: lg b is decimal logarithm. If the logarithm has the number e as its base, then write the expression: ln b – natural logarithm. It is understood that the result of any is the power to which the base number must be raised to obtain the number b.
When finding the sum of two functions, you simply need to differentiate them one by one and add the results: (u+v)" = u"+v";
When finding the derivative of the product of two functions, it is necessary to multiply the derivative of the first function by the second and add the derivative of the second function multiplied by the first function: (u*v)" = u"*v+v"*u;
In order to find the derivative of the quotient of two functions, it is necessary to subtract from the product of the derivative of the dividend multiplied by the divisor function the product of the derivative of the divisor multiplied by the function of the dividend, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;
If given complex function, then it is necessary to multiply the derivative of internal function and the derivative of the external one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).
Using the results obtained above, you can differentiate almost any function. So let's look at a few examples:
y=x^4, y"=4*x^(4-1)=4*x^3;
y=2*x^3*(e^x-x^2+6), y"=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2 *x));
There are also problems involving calculating the derivative at a point. Let the function y=e^(x^2+6x+5) be given, you need to find the value of the function at the point x=1.
1) Find the derivative of the function: y"=e^(x^2-6x+5)*(2*x +6).
2) Calculate the value of the function in given point y"(1)=8*e^0=8
Video on the topic
Learn the table of elementary derivatives. This will significantly save time.
Sources:
- derivative of a constant
So, what's the difference? ir rational equation from the rational? If the unknown variable is under the sign square root, then the equation is considered irrational.
Instructions
The main method for solving such equations is the method of constructing both sides equations into a square. However. this is natural, the first thing you need to do is get rid of the sign. This method is not technically difficult, but sometimes it can lead to trouble. For example, the equation is v(2x-5)=v(4x-7). By squaring both sides you get 2x-5=4x-7. Solving such an equation is not difficult; x=1. But the number 1 will not be given equations. Why? Substitute one into the equation instead of the value of x. And the right and left sides will contain expressions that do not make sense, that is. This value is not valid for a square root. Therefore 1 is an extraneous root, and therefore given equation has no roots.
So, an irrational equation is solved using the method of squaring both its sides. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots into the original equation.
Consider another one.
2х+vх-3=0
Of course, this equation can be solved using the same equation as the previous one. Move Compounds equations, which do not have a square root, to the right side and then use the squaring method. solve the resulting rational equation and roots. But also another, more elegant one. Enter a new variable; vх=y. Accordingly, you will receive an equation of the form 2y2+y-3=0. That is, the usual quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vх=1; vх=-3/2. The second equation has no roots; from the first we find that x=1. Don't forget to check the roots.
Solving identities is quite simple. To do this you need to do identity transformations until the goal is achieved. Thus, with the help of the simplest arithmetic operations the task at hand will be solved.
You will need
- - paper;
- - pen.
Instructions
The simplest of such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), difference of squares, sum (difference), cube of the sum (difference)). In addition, there are many and trigonometric formulas, which are essentially the same identities.
Indeed, the square of the sum of two terms equal to square the first plus double the product of the first by the second and plus the square of the second, that is (a+b)^2= (a+b)(a+b)=a^2+ab +ba+b^2=a^2+2ab +b^2.
Simplify both
General principles of the solution
Repeat according to the textbook mathematical analysis or higher mathematics, which is a definite integral. As is known, the solution definite integral there is a function whose derivative gives an integrand. This function is called an antiderivative. By this principle and constructs the main integrals.Determine by the form of the integrand which of the table integrals fits in in this case. It is not always possible to determine this immediately. Often, the tabular form becomes noticeable only after several transformations to simplify the integrand.
Variable Replacement Method
If the integrand function is trigonometric function, whose argument contains some polynomial, then try using the variable replacement method. In order to do this, replace the polynomial in the argument of the integrand with some new variable. Based on the relationship between the new and old variables, determine the new limits of integration. Differentiation given expression find a new differential in . So you will get the new kind of the previous integral, close to or even corresponding to any tabular one.Solving integrals of the second kind
If the integral is an integral of the second kind, a vector form of the integrand, then you will need to use the rules for the transition from these integrals to scalar ones. One such rule is the Ostrogradsky-Gauss relation. This law allows you to go from the rotor flux of some vector function to the triple integral over the divergence of a given vector field.Substitution of integration limits
After finding the antiderivative, it is necessary to substitute the limits of integration. First, substitute the value of the upper limit into the expression for the antiderivative. You will get some number. Next, subtract from the resulting number another number obtained from the lower limit into the antiderivative. If one of the limits of integration is infinity, then when substituting it into antiderivative function it is necessary to go to the limit and find what the expression strives for.If the integral is two-dimensional or three-dimensional, then you will have to represent the limits of integration geometrically to understand how to evaluate the integral. Indeed, in the case of, say, a three-dimensional integral, the limits of integration can be entire planes that limit the volume being integrated.