As you know, when multiplying expressions with degrees, their indicators always add up (a b * a c = a b + c ). This mathematical law was deduced by Archimedes, and later, in the VIII century, the mathematician Virasen created a table of integer indices. It was they who served for the further discovery of the logarithms. Examples of using this function can be found almost everywhere where it is necessary to simplify the cumbersome multiplication by simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. Simple and affordable language.
Definition in math
A logarithm is an expression of the following form: log a b = c, that is, the logarithm of any non-negative number (that is, any positive) "b" based on its base "a" is the degree of "c" to which the base "a" must be raised so that finally get the value of "b". Let's analyze the logarithm with examples, let's say there is an expression log 2 8. How to find the answer? Very simple, you need to find such a degree that from 2 to the desired degree you get 8. Having done some calculations in your mind, we get the number 3! And rightly so, because 2 in degree 3 gives the number 8 in the answer.
Varieties of Logarithms
For many pupils and students this topic seems complicated and incomprehensible, but in fact the logarithms are not so scary, the main thing is to understand their general meaning and remember their properties and some rules. There are three separate types of logarithmic expressions:
- The natural logarithm of ln a, where the base is the Euler number (e = 2.7).
- The decimal logarithm is log a, where the base is the number 10.
- The logarithm of any number b at the base a> 1.
Each of them is solved in a standard way, including simplification, reduction and subsequent reduction to a single logarithm using logarithmic theorems. To obtain the correct values of the logarithms, one should remember their properties and the sequence of actions when solving them.
Rules and some restrictions
In mathematics, there are several rules, restrictions, which are accepted as an axiom, that is, are not subject to discussion and are true. For example, it is impossible to divide the numbers by zero, and it is still impossible to extract the root of an even degree from negative numbers. Logarithms also have their own rules, following which you can easily learn to work even with long and capacious logarithmic expressions:
- the base "a" should always be greater than zero, and at the same time not be equal to 1, otherwise the expression will lose its meaning, because "1" and "0" are always equal in any degree to their values;
- if a> 0, then a b > 0, it turns out that "c" must be greater than zero.
How to solve logarithms?
For example, the task is given to find the answer to the equation 10 x = 100. It is very easy, you need to choose such a degree, raising to which the number ten, we get 100. This, of course, is a quadratic degree! 10 2 = 100.
Now let's imagine this expression as a logarithmic. We get log 10 100 = 2. When solving the logarithms, all actions practically converge to find the degree to which you need to enter the base of the logarithm to get the given number.
To accurately determine the value of an unknown degree, you must learn to work with a table of degrees. It looks like this:
As you can see, some degree indicators can be guessed intuitively if there is a technical mentality and knowledge of the multiplication table. However, for large values, a degree table is required. It can be used even by those who do not understand anything at all in complex mathematical topics. The left column shows the numbers (base a), the top row of numbers is the value of degree c to which the number a is raised. At the intersection, the values of numbers that are the answer (a c = b) are defined in the cells. Take, for example, the very first cell with the number 10 and square it, we get the value 100, which is indicated at the intersection of our two cells. Everything is so simple and easy that even the real humanities will understand!
Equations and Inequalities
It turns out that under certain conditions the exponent is the logarithm. Therefore, any mathematical numerical expressions can be written in the form of a logarithmic equality. For example, 3 4 = 81 can be written as the logarithm of 81 on base 3, which is four (log 3 81 = 4). For negative degrees, the rules are the same: 2 -5 = 1/32 we write in the form of a logarithm, we get log 2 (1/32) = -5. One of the most fascinating sections of mathematics is the theme of "logarithms." We will consider examples and solutions of equations just below, immediately after studying their properties. Now let's look at how inequalities look and how to distinguish them from equations.
An expression of the following form is given: log 2 (x-1)> 3 - it is a logarithmic inequality, since the unknown value of "x" is under the sign of the logarithm. And also in the expression two quantities are compared: the logarithm of the desired number on the basis of two is greater than the number three.
The most important difference between logarithmic equations and inequalities is that equations with logarithms (an example is the logarithm of 2 x = √9) they imply one or more specific numerical values in the answer, while when solving inequalities, both the range of admissible values and the break points of this function are determined. As a result, the answer is not a simple set of individual numbers as in the answer of the equation, but a continuous series or set of numbers.
Basic logarithm theorems
When solving primitive tasks on finding the values of the logarithm, its properties may not be known. However, when it comes to logarithmic equations or inequalities, first of all, it is necessary to clearly understand and put into practice all the basic properties of logarithms. We will get acquainted with examples of equations later, let's first analyze each property in more detail.
- The main identity looks like this: logaB = B. It applies only when a is greater than 0, not equal to one, and B is greater than zero.
- The logarithm of the product can be represented in the following formula: log d (s 1 * s 2) = log d s 1 + log d s 2. Moreover, a prerequisite is: d, s 1 and s 2 > 0; a ≠ 1. We can give a proof for this formula of logarithms, with examples and a solution. Let log a s 1 = f 1 and log a s 2 = f 2 , then a f1 = s 1 , a f2 = s 2. We get that s 1 * s 2 = a f1 * a f2 = a f1 + f2 ( properties of degrees), and then by definition: log a (s 1 * s 2 ) = f 1 + f 2 = log a s1 + log a s 2, as required.
- The logarithm of the quotient looks like this: log a (s 1 / s 2 ) = log a s 1 - log a s 2.
- A theorem in the form of a formula takes the following form: log a q b n = n / q log a b.
This formula is called the "property of the degree of the logarithm." It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics is based on regular postulates. Let's look at the proof.
Let log a b = t, it turns out a t = b. If both parts are raised to the power m: a tn = b n ;
but since a tn = (a q ) nt / q = b n , therefore, log a q b n = (n * t) / t, then log a q b n = n / q log a b. The theorem is proved.
Examples of problems and inequalities
The most common types of problems on the topic of logarithms are examples of equations and inequalities. They are found in almost all problem books, and are also included in the required part of math exams. To enter the university or take entrance examinations in mathematics, you need to know how to correctly solve such tasks.
Unfortunately, there is no single plan or scheme for solving and determining the unknown value of the logarithm, however, certain rules can be applied to each mathematical inequality or logarithmic equation. First of all, you need to find out whether it is possible to simplify the expression or lead to a general view. Long logarithmic expressions can be simplified if their properties are used correctly. Let's get to know them soon.
When solving the logarithmic equations, it is necessary to determine what kind of logarithm is in front of us: an example of an expression may contain a natural logarithm or a decimal.
Here are examples of decimal logarithms: ln100, ln1026. Their solution comes down to the fact that it is necessary to determine the degree to which the base 10 will be equal to 100 and 1026, respectively. For solutions of natural logarithms, one needs to apply logarithmic identities or their properties. Let's look at examples of solving logarithmic problems of various types.
How to use logarithm formulas: with examples and solutions
So, let's look at examples of using the basic theorems on logarithms.
- The property of the product logarithm can be used in tasks where it is necessary to decompose the large value of the number b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4 * 128) = log 2 512. The answer is 9.
- log 4 8 = log 2 2 2 3 = 3/2 log 2 2 = 1.5 - as you can see, using the fourth property of the degree of the logarithm, it was possible to solve at first glance a complex and unsolvable expression. It is only necessary to factor the basis and then derive the degree from the sign of the logarithm.
Tasks from the exam
Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the exam (state exam for all school graduates). Usually these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most difficult and voluminous tasks). The exam involves an accurate and perfect knowledge of the topic "Natural Logarithms".
Examples and solutions to problems are taken from the official exams. Let's see how such tasks are solved.
Given log 2 (2x-1) = 4. Solution:
we rewrite the expression, simplifying it a bit log 2 (2x-1) = 2 2 , by the definition of the logarithm we get that 2x-1 = 2 4 , therefore 2x = 17; x = 8.5.
Below are a few recommendations, following which you can easily solve all equations containing expressions that are under the sign of the logarithm.
- All logarithms are best reduced to the same base so that the solution is not cumbersome and confusing.
- The whole expression under the logarithm sign is indicated as positive, therefore, when the factor makes the exponent of the expression, which stands under the sign of the logarithm and as its basis, the expression remaining under the logarithm should be positive.