Today we will talk about logarithm formulas and give demonstration solution examples.

By themselves, they imply solution patterns according to the basic properties of logarithms. Before applying the logarithm formulas to the solution, we recall for you, first all the properties:

Now, based on these formulas (properties), we show examples of solving logarithms.

Examples of solving logarithms based on formulas.

Logarithm a positive number b in base a (denoted log a b) is the exponent to which a must be raised to get b, with b > 0, a > 0, and 1.

According to the definition log a b = x, which is equivalent to a x = b, so log a a x = x.

Logarithms, examples:

log 2 8 = 3, because 2 3 = 8

log 7 49 = 2 because 7 2 = 49

log 5 1/5 = -1, because 5 -1 = 1/5

Decimal logarithm is an ordinary logarithm, the base of which is 10. Denoted as lg.

log 10 100 = 2 because 10 2 = 100

natural logarithm- also the usual logarithm logarithm, but with the base e (e \u003d 2.71828 ... - an irrational number). Referred to as ln.

It is desirable to remember the formulas or properties of logarithms, because we will need them later when solving logarithms, logarithmic equations and inequalities. Let's work through each formula again with examples.

• Basic logarithmic identity
  a log a b = b

  8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9

• The logarithm of the product is equal to the sum of the logarithms
  log a (bc) = log a b + log a c

  log 3 8.1 + log 3 10 = log 3 (8.1*10) = log 3 81 = 4

• The logarithm of the quotient is equal to the difference of the logarithms
  log a (b/c) = log a b - log a c

  9 log 5 50 /9 log 5 2 = 9 log 5 50- log 5 2 = 9 log 5 25 = 9 2 = 81

• Properties of the degree of a logarithmable number and the base of the logarithm

  The exponent of a logarithm number log a b m = mlog a b

  Exponent of the base of the logarithm log a n b =1/n*log a b

  log a n b m = m/n*log a b,

  if m = n, we get log a n b n = log a b

  log 4 9 = log 2 2 3 2 = log 2 3

• Transition to a new foundation
  log a b = log c b / log c a,

  if c = b, we get log b b = 1

  then log a b = 1/log b a

  log 0.8 3*log 3 1.25 = log 0.8 3*log 0.8 1.25/log 0.8 3 = log 0.8 1.25 = log 4/5 5/4 = -1

As you can see, the logarithm formulas are not as complicated as they seem. Now, having considered examples of solving logarithms, we can move on to logarithmic equations. We will consider examples of solving logarithmic equations in more detail in the article: "". Do not miss!

If you still have questions about the solution, write them in the comments to the article.

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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 made it possible to replace complex operations with 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 were compiled 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 equal to the sum of the 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 for the 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. For a long time, engineers used the so-called graph paper for these purposes.

In the 17th century, the first auxiliary analog computing conditions appeared, which by the 19th century had acquired a finished form. 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 only be positive if both 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 removed from real life that the logarithm has suddenly gained a lot of importance in describing objects in the real world. It is difficult to find a science where it is not used. This fully applies not only to the natural, but also to the humanities fields of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical research methods 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 stimulus intensity value to the lower intensity value.

After the above examples, it is no longer surprising that the theme of logarithms is also widely used in biology. Whole volumes can be written about biological forms corresponding to logarithmic spirals.

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.

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What is a logarithm?

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For those who strongly "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? Good. Now, for some 10 - 20 minutes you:

1. Understand what is a logarithm.

2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.

3. Learn to calculate simple logarithms.

Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...

I feel you doubt ... Well, keep time! Go!

First, solve the following equation in your mind:

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. Learning - with interest!)

you can get acquainted with functions and derivatives.

Let's start with properties of the logarithm of unity. Its formulation is as follows: the logarithm of unity is equal to zero, that is, log a 1=0 for any a>0 , a≠1 . The proof is straightforward: since a 0 =1 for any a that satisfies the above conditions a>0 and a≠1 , then the proven equality log a 1=0 immediately follows from the definition of the logarithm.

Let's give examples of application of the considered property: log 3 1=0 , lg1=0 and .

Let's move on to the next property: the logarithm of a number equal to the base is equal to one, i.e, log a a=1 for a>0 , a≠1 . Indeed, since a 1 =a for any a , then by the definition of the logarithm log a a=1 .

Examples of using this property of logarithms are log 5 5=1 , log 5.6 5.6 and lne=1 .

For example, log 2 2 7 =7 , log10 -4 =-4 and .

Logarithm of the product of two positive numbers x and y is equal to the product of the logarithms of these numbers: log a (x y)=log a x+log a y, a>0 , a≠1 . Let us prove the property of the logarithm of the product. Due to the properties of the degree a log a x+log a y =a log a x a log a y, and since by the main logarithmic identity a log a x =x and a log a y =y , then a log a x a log a y =x y . Thus, a log a x+log a y =x y , whence the required equality follows by the definition of the logarithm.

Let's show examples of using the property of the logarithm of the product: log 5 (2 3)=log 5 2+log 5 3 and .

The product logarithm property can be generalized to the product of a finite number n of positive numbers x 1 , x 2 , …, x n as log a (x 1 x 2 ... x n)= log a x 1 + log a x 2 +…+ log a x n . This equality is easily proved.

For example, the natural logarithm of a product can be replaced by the sum of three natural logarithms of the numbers 4 , e , and .

Logarithm of the quotient of two positive numbers x and y is equal to the difference between the logarithms of these numbers. The quotient logarithm property corresponds to a formula of the form , where a>0 , a≠1 , x and y are some positive numbers. The validity of this formula is proved like the formula for the logarithm of the product: since , then by the definition of the logarithm .

Here is an example of using this property of the logarithm: .

Let's move on to property of the logarithm of degree. The logarithm of a degree is equal to the product of the exponent and the logarithm of the modulus of the base of this degree. We write this property of the logarithm of the degree in the form of a formula: log a b p =p log a |b|, where a>0 , a≠1 , b and p are numbers such that the degree of b p makes sense and b p >0 .

We first prove this property for positive b . The basic logarithmic identity allows us to represent the number b as a log a b , then b p =(a log a b) p , and the resulting expression, due to the power property, is equal to a p log a b . So we arrive at the equality b p =a p log a b , from which, by the definition of the logarithm, we conclude that log a b p =p log a b .

It remains to prove this property for negative b . Here we note that the expression log a b p for negative b makes sense only for even exponents p (since the value of the degree b p must be greater than zero, otherwise the logarithm will not make sense), and in this case b p =|b| p . Then b p =|b| p =(a log a |b|) p =a p log a |b|, whence log a b p =p log a |b| .

For example, and ln(-3) 4 =4 ln|-3|=4 ln3 .

It follows from the previous property property of the logarithm from the root: the logarithm of the root of the nth degree is equal to the product of the fraction 1/n and the logarithm of the root expression, that is, , where a>0 , a≠1 , n is a natural number greater than one, b>0 .

The proof is based on the equality (see ), which is valid for any positive b , and the property of the logarithm of the degree: .

Here is an example of using this property: .

Now let's prove conversion formula to the new base of the logarithm kind . To do this, it suffices to prove the validity of the equality log c b=log a b log c a . The basic logarithmic identity allows us to represent the number b as a log a b , then log c b=log c a log a b . It remains to use the property of the logarithm of the degree: log c a log a b = log a b log c a. Thus, the equality log c b=log a b log c a is proved, which means that the formula for the transition to a new base of the logarithm is also proved.

Let's show a couple of examples of applying this property of logarithms: and .

The formula for moving to a new base allows you to move on to working with logarithms that have a “convenient” base. For example, it can be used to switch to natural or decimal logarithms so that you can calculate the value of the logarithm from the table of logarithms. The formula for the transition to a new base of the logarithm also allows in some cases to find the value of a given logarithm, when the values ​​of some logarithms with other bases are known.

Often used is a special case of the formula for the transition to a new base of the logarithm for c=b of the form . This shows that log a b and log b a – . For example, .

Also often used is the formula , which is useful for finding logarithm values. To confirm our words, we will show how the value of the logarithm of the form is calculated using it. We have . To prove the formula it is enough to use the transition formula to the new base of the logarithm a: .

It remains to prove the comparison properties of logarithms.

Let us prove that for any positive numbers b 1 and b 2 , b 1 log a b 2 , and for a>1, the inequality log a b 1

Finally, it remains to prove the last of the listed properties of logarithms. We confine ourselves to proving its first part, that is, we prove that if a 1 >1 , a 2 >1 and a 1 1 is true log a 1 b>log a 2 b . The remaining statements of this property of logarithms are proved by a similar principle.

Let's use the opposite method. Suppose that for a 1 >1 , a 2 >1 and a 1 1 log a 1 b≤log a 2 b is true. By the properties of logarithms, these inequalities can be rewritten as and respectively, and from them it follows that log b a 1 ≤log b a 2 and log b a 1 ≥log b a 2, respectively. Then, by the properties of powers with the same bases, the equalities b log b a 1 ≥b log b a 2 and b log b a 1 ≥b log b a 2 must be satisfied, that is, a 1 ≥a 2 . Thus, we have arrived at a contradiction to the condition a 1

Bibliography.

• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).