Decimal logarithm of numbers how to solve. Assignments from the exam

Logarithm of b (b> 0) to base a (a> 0, a ≠ 1) Is the exponent to which you need to raise the number a to get b.

The logarithm of b to base 10 can be written as lg (b), and the logarithm to base e (natural logarithm) is ln (b).

Often used when solving problems with logarithms:

Properties of logarithms

There are four main properties of logarithms.

Let a> 0, a ≠ 1, x> 0, and y> 0.

Property 1. Logarithm of the product

Logarithm of the product is equal to the sum of the logarithms:

log a (x ⋅ y) = log a x + log a y

Property 2. Logarithm of the quotient

Logarithm of the quotient is equal to the difference of logarithms:

log a (x / y) = log a x - log a y

Property 3. Logarithm of the degree

Logarithm of degree is equal to the product degrees per logarithm:

If the base of the logarithm is in power, then another formula works:

Property 4. Logarithm of the root

This property can be obtained from the property of the logarithm of the degree, since the root of the nth degree is equal to the degree 1 / n:

The formula for the transition from a logarithm in one base to a logarithm in another base

This formula is also often used to solve various problems for logarithms:

A special case:

Comparison of logarithms (inequalities)

Suppose we have 2 functions f (x) and g (x) under logarithms with the same bases and there is an inequality sign between them:

To compare them, you first need to look at the base of the logarithms of a:

If a> 0, then f (x)> g (x)> 0
If 0< a < 1, то 0 < f(x) < g(x)

How to solve problems with logarithms: examples

Logarithm tasks included in the USE in mathematics for grade 11 in task 5 and task 7, you can find tasks with solutions on our website in the relevant sections. Also, tasks with logarithms are found in the bank of tasks in mathematics. All examples can be found through the site search.

What is a logarithm

Logarithms have always been considered a difficult topic in high school maths. There are many different definitions logarithm, but most textbooks somehow use the most difficult and unfortunate ones.

We will define the logarithm simply and clearly. To do this, let's create a table:

So, we have before us powers of two.

Logarithms - properties, formulas, how to solve

If you take the number from the bottom line, then you can easily find the degree to which you have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.

And now - actually, the definition of the logarithm:

base a from argument x is the power to which the number a must be raised to get the number x.

Notation: log a x = b, where a is the base, x is the argument, b is actually what the logarithm is.

For example, 2 3 = 8 ⇒log 2 8 = 3 (log base 2 of 8 is three, since 2 3 = 8). With the same success log 2 64 = 6, since 2 6 = 64.

The operation of finding the logarithm of a number in a given base is called. So, let's add a new line to our table:

2 1	2 2	2 3	2 4	2 5	2 6
2	4	8	16	32	64
log 2 2 = 1	log 2 4 = 2	log 2 8 = 3	log 2 16 = 4	log 2 32 = 5	log 2 64 = 6

Unfortunately, not all logarithms are calculated so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем more degree two, the larger the number will be.

Such numbers are called irrational: the numbers after the decimal point can be written indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100.

It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many are confused about where the basis is, and where is the argument. To avoid annoying misunderstandings, just take a look at the picture:

Before us is nothing more than the definition of the logarithm. Remember: logarithm is the degree to which the base must be raised to get the argument. It is the base that is raised to the power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and no confusion arises.

How to count logarithms

We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the log sign. To begin with, we note that two important facts follow from the definition:

Argument and radix must always be greater than zero. This follows from the definition of the degree by a rational indicator, to which the definition of the logarithm is reduced.
The base must be different from one, since one is still one to any degree. Because of this, the question "to what degree one must raise the unit to get a two" is meaningless. There is no such degree!

Such restrictions are called range of valid values(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒x> 0, a> 0, a ≠ 1.

Note that there is no restriction on the number b (the value of the logarithm). For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1.

However, now we are considering only numerical expressions, where knowing the ODV of the logarithm is not required. All restrictions have already been taken into account by the task compilers. But when the logarithmic equations and inequalities come in, the DHS requirements will become mandatory. Indeed, at the base and in the argument there can be very strong constructions that do not necessarily correspond to the above restrictions.

Now consider general scheme calculating logarithms. It consists of three steps:

Present radix a and argument x as a power with the smallest possible radix greater than one. Along the way, it is better to get rid of decimal fractions;
Solve the equation for variable b: x = a b;
The resulting number b will be the answer.

That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement for the base to be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately translate them into ordinary ones, there will be many times fewer errors.

Let's see how this scheme works with specific examples:

Task. Calculate the log of: log 5 25

Let's represent the base and the argument as a power of five: 5 = 5 1; 25 = 5 2;

Let's compose and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒5 b = 5 2 ⇒ b = 2;

Received the answer: 2.

Task. Calculate the logarithm:

Task. Calculate the log of: log 4 64

Let's represent the base and the argument as a power of two: 4 = 2 2; 64 = 2 6;
Let's compose and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒2 2b = 2 6 ⇒2b = 6 ⇒ b = 3;
Received the answer: 3.

Task. Calculate the logarithm: log 16 1

Let's represent the base and the argument as a power of two: 16 = 2 4; 1 = 2 0;
Let's compose and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒2 4b = 2 0 ⇒4b = 0 ⇒ b = 0;
Received the answer: 0.

Task. Calculate the log of: log 7 14

Let's represent the base and the argument as a power of seven: 7 = 7 1; 14 is not represented as a power of seven, since 7 1< 14 < 7 2 ;
From the previous paragraph it follows that the logarithm is not counted;
The answer is no change: log 7 14.

A small note on the last example. How do you ensure that a number is not an exact power of another number? It's very simple - just factor it into prime factors. If the factorization contains at least two different factors, the number is not an exact power.

Task. Find out if the exact powers of the number are: 8; 48; 81; 35; 14.

8 = 2 2 2 = 2 3 - the exact degree, because there is only one factor;
48 = 6 · 8 = 3 · 2 · 2 · 2 · 2 = 3 · 2 4 - is not an exact degree, since there are two factors: 3 and 2;
81 = 9 9 = 3 3 3 3 3 = 3 4 - exact degree;
35 = 7 · 5 - again not an exact degree;
14 = 7 2 - again not an exact degree;

Note also that the primes themselves are always exact powers of themselves.

Decimal logarithm

Some logarithms are so common that they have special name and designation.

of argument x is the base 10 logarithm, i.e. the power to which the number 10 must be raised to get the number x. Designation: lg x.

For example, lg 10 = 1; lg 100 = 2; lg 1000 = 3 - etc.

From now on, when a phrase like "Find lg 0.01" appears in a textbook, you should know: this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x

Everything that is true for ordinary logarithms is also true for decimals.

Natural logarithm

There is another logarithm that has its own notation. In a way, it is even more important than decimal. It is about the natural logarithm.

of argument x is the logarithm base e, i.e. the power to which the number e must be raised to get the number x. Designation: ln x.

Many will ask: what else is the number e? This is an irrational number, its exact value it is impossible to find and record. I will give only its first figures:
e = 2.718281828459 ...

We will not delve into what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x

Thus, ln e = 1; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, units: ln 1 = 0.

For natural logarithms all the rules are true that are true for ordinary logarithms.

Logarithm. Properties of the logarithm (power of the logarithm).

How do I represent a number as a logarithm?

We use the definition of a logarithm.

The logarithm is an exponent to which the base must be raised to get the number under the sign of the logarithm.

Thus, in order to represent some number c in the form of a logarithm to the base a, it is necessary to put the power with the same base as the base of the logarithm under the sign of the logarithm, and write this number c in the exponent:

In the form of a logarithm, absolutely any number can be represented - positive, negative, whole, fractional, rational, irrational:

In order not to confuse a and c under stressful conditions of a control or exam, you can use the following rule to memorize:

what is below goes down, what is above goes up.

For example, you might want to represent the number 2 as a logarithm to base 3.

We have two numbers - 2 and 3. These numbers are the base and the exponent, which we will write under the sign of the logarithm. It remains to determine which of these numbers needs to be written down, in the base of the degree, and which - up, in the exponent.

The base 3 in the logarithm is at the bottom, which means that when we represent two as a logarithm to the base 3, 3 will also be written down to the base.

2 stands above the three. And in the notation of the power of two, we write it above the three, that is, in the exponent:

Logarithms. First level.

Logarithms

Logarithm positive number b by reason a, where a> 0, a ≠ 1, is called the exponent to which the number must be raised a, To obtain b.

Definition of the logarithm can be briefly written like this:

This equality is valid for b> 0, a> 0, a ≠ 1. It is usually called logarithmic identity.
The action of finding the logarithm of a number is called by taking the logarithm.

Logarithm properties:

Logarithm of the product:

Logarithm of the quotient of division:

Replacing the base of the logarithm:

Logarithm of the degree:

Logarithm of the root:

Power logarithm:

Decimal and natural logarithms.

Decimal logarithm numbers call the base 10 logarithm of this number and write & nbsp lg b
Natural logarithm numbers call the base logarithm of that number e, where e- an irrational number, approximately equal to 2.7. In this case, they write ln b.

Other notes on algebra and geometry

Basic properties of logarithms

Logarithms, like any numbers, can be added, subtracted and transformed in every way. But since logarithms are not exactly ordinary numbers, there are rules here, which are called basic properties.

It is imperative to know these rules - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.

Addition and subtraction of logarithms

Consider two logarithms with the same bases: log a x and log a y. Then they can be added and subtracted, and:

log a x + log a y = log a (x y);
log a x - log a y = log a (x: y).

So, the sum of the logarithms is equal to the logarithm of the product, and the difference is the logarithm of the quotient. Please note, the key point here is - identical grounds... If the reasons are different, these rules do not work!

These formulas will help you calculate a logarithmic expression even when its individual parts are not counted (see the lesson "What is a logarithm"). Take a look at the examples - and see:

Log 6 4 + log 6 9.

Since the bases of the logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.

Task. Find the value of the expression: log 2 48 - log 2 3.

The bases are the same, we use the difference formula:
log 2 48 - log 2 3 = log 2 (48: 3) = log 2 16 = 4.

Task. Find the value of the expression: log 3 135 - log 3 5.

Again the bases are the same, so we have:
log 3 135 - log 3 5 = log 3 (135: 5) = log 3 27 = 3.

As you can see, the original expressions are composed of "bad" logarithms, which are not separately counted. But after transformations, quite normal numbers are obtained. Many are built on this fact. test papers... But what control - such expressions in all seriousness (sometimes - practically unchanged) are offered on the exam.

Removing the exponent from the logarithm

Now let's complicate the task a little. What if the base or argument of the logarithm is based on a degree? Then the exponent of this degree can be taken out of the sign of the logarithm according to the following rules:

It's easy to see that the last rule follows the first two. But it's better to remember it all the same - in some cases it will significantly reduce the amount of computation.

Of course, all these rules make sense if the ODL of the logarithm is observed: a> 0, a ≠ 1, x> 0. And one more thing: learn to apply all formulas not only from left to right, but also vice versa, i.e. you can enter the numbers in front of the sign of the logarithm into the logarithm itself.

How to solve logarithms

This is what is most often required.

Task. Find the value of the expression: log 7 49 6.

Let's get rid of the degree in the argument using the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12

Task. Find the meaning of the expression:

Note that the denominator contains the logarithm, the base and argument of which are exact powers: 16 = 2 4; 49 = 7 2. We have:

I think the last example needs some clarification. Where did the logarithms disappear? Until the very last moment, we work only with the denominator. We presented the base and the argument of the logarithm standing there in the form of degrees and brought out the indicators - we got a "three-story" fraction.

Now let's look at the basic fraction. The numerator and denominator contain the same number: log 2 7. Since log 2 7 ≠ 0, we can cancel the fraction - the denominator remains 2/4. According to the rules of arithmetic, the four can be transferred to the numerator, which was done. The result was the answer: 2.

Moving to a new foundation

Speaking about the rules for addition and subtraction of logarithms, I specifically emphasized that they only work for the same bases. What if the reasons are different? What if they are not exact powers of the same number?

Formulas for the transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem:

Let the logarithm log a x be given. Then, for any number c such that c> 0 and c ≠ 1, the following equality holds:

In particular, if we put c = x, we get:

From the second formula it follows that it is possible to swap the base and the argument of the logarithm, but in this case the whole expression is "reversed", i.e. the logarithm appears in the denominator.

These formulas are rarely found in conventional numeric expressions. It is possible to assess how convenient they are only when deciding logarithmic equations and inequalities.

However, there are tasks that are generally not solved except by the transition to a new foundation. Consider a couple of these:

Task. Find the value of the expression: log 5 16 log 2 25.

Note that the arguments of both logarithms contain exact degrees. Let's take out the indicators: log 5 16 = log 5 2 4 = 4log 5 2; log 2 25 = log 2 5 2 = 2 log 2 5;

Now let's "flip" the second logarithm:

Since the product does not change from the permutation of the factors, we calmly multiplied the four and two, and then dealt with the logarithms.

Task. Find the value of the expression: log 9 100 · lg 3.

The base and argument of the first logarithm are exact degrees. Let's write this down and get rid of the metrics:

Now let's get rid of the decimal logarithm by moving to the new base:

Basic logarithmic identity

Often in the process of solving it is required to represent a number as a logarithm to a given base.

In this case, the formulas will help us:

In the first case, the number n becomes the exponent in the argument. The number n can be absolutely anything, because it is just the value of the logarithm.

The second formula is actually a paraphrased definition. It is called that:.

Indeed, what happens if the number b is raised to such a power that the number b to this power gives the number a? That's right: you get this very number a. Read this paragraph carefully again - many people "hang" on it.

Like the formulas for transition to a new base, the basic logarithmic identity is sometimes the only possible solution.

Task. Find the meaning of the expression:

Note that log 25 64 = log 5 8 - just moved the square out of the base and the logarithm argument. Taking into account the rules for multiplying degrees with the same base, we get:

If someone is not in the know, it was a real problem from the exam 🙂

Logarithmic unit and logarithmic zero

In conclusion, I will give two identities that can hardly be called properties - rather, they are consequences of the definition of the logarithm. They are constantly encountered in problems and, surprisingly, create problems even for "advanced" students.

log a a = 1 is. Remember once and for all: the logarithm to any base a from this base is equal to one.
log a 1 = 0 is. The base a can be anything, but if the argument contains one - the logarithm is zero! Because a 0 = 1 is a direct consequence of the definition.

That's all the properties. Be sure to practice putting them into practice! Download the cheat sheet at the beginning of the lesson, print it out, and solve the problems.

basic properties.

logax + logay = loga (x y);
logax - logay = loga (x: y).

identical grounds

Log6 4 + log6 9.

Now let's complicate the task a little.

Examples of solving logarithms

What if the base or argument of the logarithm is based on a degree? Then the exponent of this degree can be taken out of the sign of the logarithm according to the following rules:

Of course, all these rules make sense if the ODL of the logarithm is observed: a> 0, a ≠ 1, x>

Task. Find the meaning of the expression:

Moving to a new foundation

Let the logarithm be given. Then, for any number c such that c> 0 and c ≠ 1, the following equality holds:

Task. Find the meaning of the expression:

Basic properties of the logarithm

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

The exponent is 2.718281828…. To remember the exponent, you can study the rule: the exponent is 2.7 and twice the year of birth of Leo Nikolaevich Tolstoy.

Basic properties of logarithms

Knowing this rule, you will know both the exact value of the exponent and the date of birth of Leo Tolstoy.

Examples for logarithms

Logarithm expressions

Example 1.
a). x = 10ac ^ 2 (a> 0, c> 0).

By properties 3.5 we calculate

4. where .

Example 2. Find x if

Example 3. Let the value of the logarithms be given

Evaluate log (x) if

Basic properties of logarithms

It is imperative to know these rules - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.

Addition and subtraction of logarithms

Consider two logarithms with the same bases: logax and logay. Then they can be added and subtracted, and:

logax + logay = loga (x y);
logax - logay = loga (x: y).

These formulas will help you calculate a logarithmic expression even when its individual parts are not counted (see the lesson "What is a logarithm"). Take a look at the examples - and see:

Since the bases of the logarithms are the same, we use the sum formula:
log6 4 + log6 9 = log6 (4 9) = log6 36 = 2.

Task. Find the value of the expression: log2 48 - log2 3.

The bases are the same, we use the difference formula:
log2 48 - log2 3 = log2 (48: 3) = log2 16 = 4.

Task. Find the value of the expression: log3 135 - log3 5.

Again the bases are the same, so we have:
log3 135 - log3 5 = log3 (135: 5) = log3 27 = 3.

As you can see, the original expressions are composed of "bad" logarithms, which are not separately counted. But after transformations, quite normal numbers are obtained. Many tests are based on this fact. But what control - such expressions in all seriousness (sometimes - practically unchanged) are offered on the exam.

Removing the exponent from the logarithm

It's easy to see that the last rule follows the first two. But it's better to remember it all the same - in some cases it will significantly reduce the amount of computation.

Task. Find the value of the expression: log7 496.

Let's get rid of the degree in the argument using the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the meaning of the expression:

Note that the denominator contains the logarithm, the base and argument of which are exact powers: 16 = 24; 49 = 72. We have:

I think the last example needs some clarification. Where did the logarithms disappear? Until the very last moment, we work only with the denominator.

Formulas for logarithms. Logarithms are examples of solutions.

We presented the base and the argument of the logarithm standing there in the form of degrees and brought out the indicators - we got a "three-story" fraction.

Now let's look at the basic fraction. The numerator and denominator contain the same number: log2 7. Since log2 7 ≠ 0, we can cancel the fraction - 2/4 will remain in the denominator. According to the rules of arithmetic, the four can be transferred to the numerator, which was done. The result was the answer: 2.

Moving to a new foundation

Formulas for the transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem:

Let the logarithm be given. Then, for any number c such that c> 0 and c ≠ 1, the following equality holds:

In particular, if we put c = x, we get:

These formulas are rarely found in conventional numeric expressions. It is possible to assess how convenient they are only when solving logarithmic equations and inequalities.

However, there are tasks that are generally not solved except by the transition to a new foundation. Consider a couple of these:

Task. Find the value of the expression: log5 16 log2 25.

Note that the arguments of both logarithms contain exact degrees. Let's take out the indicators: log5 16 = log5 24 = 4log5 2; log2 25 = log2 52 = 2log2 5;

Now let's "flip" the second logarithm:

Since the product does not change from the permutation of the factors, we calmly multiplied the four and two, and then dealt with the logarithms.

Task. Find the value of the expression: log9 100 · lg 3.

The base and argument of the first logarithm are exact degrees. Let's write this down and get rid of the metrics:

Now let's get rid of the decimal logarithm by moving to the new base:

Basic logarithmic identity

Often in the process of solving it is required to represent a number as a logarithm to a given base. In this case, the formulas will help us:

In the first case, the number n becomes the exponent in the argument. The number n can be absolutely anything, because it is just the value of the logarithm.

The second formula is actually a paraphrased definition. It is called that:.

Like the formulas for transition to a new base, the basic logarithmic identity is sometimes the only possible solution.

Task. Find the meaning of the expression:

Note that log25 64 = log5 8 - just moved the square out of the base and the logarithm argument. Taking into account the rules for multiplying degrees with the same base, we get:

If someone is not in the know, it was a real problem from the exam 🙂

Logarithmic unit and logarithmic zero

logaa = 1 is. Remember once and for all: the logarithm to any base a from this base is equal to one.
loga 1 = 0 is. The base a can be anything, but if the argument is one, the logarithm is zero! Because a0 = 1 is a direct consequence of the definition.

That's all the properties. Be sure to practice putting them into practice! Download the cheat sheet at the beginning of the lesson, print it out, and solve the problems.

Basic properties of the logarithm

The above properties need to be known, since, on their basis, almost all problems and examples are associated with logarithms are solved. The rest of the exotic properties can be deduced by mathematical manipulations with these formulas

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

When calculating the formulas for the sum and difference of logarithms (3.4) are encountered quite often. The rest are somewhat complex, but in a number of tasks they are indispensable for simplifying complex expressions and calculating their values.

Common cases of logarithms

Some of the common logarithms are those in which the base is even ten, exponential or two.
The base ten logarithm is usually called the decimal logarithm and is simply denoted lg (x).

It can be seen from the recording that the basics are not written in the recording. For example

The natural logarithm is the logarithm based on the exponent (denoted by ln (x)).

The exponent is 2.718281828…. To remember the exponent, you can study the rule: the exponent is 2.7 and twice the year of birth of Leo Nikolaevich Tolstoy. Knowing this rule, you will know both the exact value of the exponent and the date of birth of Leo Tolstoy.

And another important base two logarithm is

The derivative of the logarithm of the function is equal to one divided by the variable

The integral or antiderivative of the logarithm is determined by the dependence

The given material is enough for you to solve a wide class of problems related to logarithms and logarithms. To assimilate the material, I will give only a few common examples from school curriculum and universities.

Examples for logarithms

Logarithm expressions

Example 1.
a). x = 10ac ^ 2 (a> 0, c> 0).

By properties 3.5 we calculate

2.
By the property of the difference of logarithms, we have

3.
Using properties 3,5 we find

4. where .

A seemingly complex expression using a number of rules is simplified to the form

Finding the values of logarithms

Example 2. Find x if

Solution. For the calculation, we apply up to the last term 5 and 13 of the properties

Substitute and grieve

Since the bases are equal, we equate the expressions

Logarithms. First level.

Let the value of the logarithms be given

Evaluate log (x) if

Solution: Let us logarithm the variable to write the logarithm through the sum of the terms

This is where the acquaintance with logarithms and their properties just begins. Practice calculations, enrich your practical skills - you will soon need this knowledge to solve logarithmic equations. Having studied the basic methods for solving such equations, we will expand your knowledge for another no less important topic- logarithmic inequalities ...

Basic properties of logarithms

It is imperative to know these rules - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.

Addition and subtraction of logarithms

Consider two logarithms with the same bases: logax and logay. Then they can be added and subtracted, and:

logax + logay = loga (x y);
logax - logay = loga (x: y).

These formulas will help you calculate a logarithmic expression even when its individual parts are not counted (see the lesson "What is a logarithm"). Take a look at the examples - and see:

Task. Find the value of the expression: log6 4 + log6 9.

Since the bases of the logarithms are the same, we use the sum formula:
log6 4 + log6 9 = log6 (4 9) = log6 36 = 2.

Task. Find the value of the expression: log2 48 - log2 3.

The bases are the same, we use the difference formula:
log2 48 - log2 3 = log2 (48: 3) = log2 16 = 4.

Task. Find the value of the expression: log3 135 - log3 5.

Again the bases are the same, so we have:
log3 135 - log3 5 = log3 (135: 5) = log3 27 = 3.

As you can see, the original expressions are composed of "bad" logarithms, which are not separately counted. But after transformations, quite normal numbers are obtained. Many tests are based on this fact. But what control - such expressions in all seriousness (sometimes - practically unchanged) are offered on the exam.

Removing the exponent from the logarithm

It's easy to see that the last rule follows the first two. But it's better to remember it all the same - in some cases it will significantly reduce the amount of computation.

How to solve logarithms

This is what is most often required.

Task. Find the value of the expression: log7 496.

Let's get rid of the degree in the argument using the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the meaning of the expression:

Note that the denominator contains the logarithm, the base and argument of which are exact powers: 16 = 24; 49 = 72. We have:

Moving to a new foundation

Formulas for the transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem:

Let the logarithm be given. Then, for any number c such that c> 0 and c ≠ 1, the following equality holds:

In particular, if we put c = x, we get:

These formulas are rarely found in conventional numeric expressions. It is possible to assess how convenient they are only when solving logarithmic equations and inequalities.

However, there are tasks that are generally not solved except by the transition to a new foundation. Consider a couple of these:

Task. Find the value of the expression: log5 16 log2 25.

Note that the arguments of both logarithms contain exact degrees. Let's take out the indicators: log5 16 = log5 24 = 4log5 2; log2 25 = log2 52 = 2log2 5;

Now let's "flip" the second logarithm:

Since the product does not change from the permutation of the factors, we calmly multiplied the four and two, and then dealt with the logarithms.

Task. Find the value of the expression: log9 100 · lg 3.

The base and argument of the first logarithm are exact degrees. Let's write this down and get rid of the metrics:

Now let's get rid of the decimal logarithm by moving to the new base:

Basic logarithmic identity

Often in the process of solving it is required to represent a number as a logarithm to a given base. In this case, the formulas will help us:

In the first case, the number n becomes the exponent in the argument. The number n can be absolutely anything, because it is just the value of the logarithm.

The second formula is actually a paraphrased definition. It is called that:.

Like the formulas for transition to a new base, the basic logarithmic identity is sometimes the only possible solution.

Task. Find the meaning of the expression:

Note that log25 64 = log5 8 - just moved the square out of the base and the logarithm argument. Taking into account the rules for multiplying degrees with the same base, we get:

If someone is not in the know, it was a real problem from the exam 🙂

Logarithmic unit and logarithmic zero

logaa = 1 is. Remember once and for all: the logarithm to any base a from this base is equal to one.
loga 1 = 0 is. The base a can be anything, but if the argument is one, the logarithm is zero! Because a0 = 1 is a direct consequence of the definition.

That's all the properties. Be sure to practice putting them into practice! Download the cheat sheet at the beginning of the lesson, print it out, and solve the problems.

What is a logarithm?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who are "very even ...")

What is a logarithm? How do you solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered difficult, incomprehensible and scary. Especially - equations with logarithms.

This is absolutely not the case. Absolutely! Don't believe me? Okay. Now, in some 10 - 20 minutes, you:

1. Understand what is logarithm.

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

3. Learn to calculate simple logarithms.

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

I feel you are in doubt ... Well, watch the time! Go!

Start by solving the following 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. Instant validation testing. Learning - with interest!)

you can get acquainted with functions and derivatives.

As society developed and production became more complex, mathematics also developed. Moving from simple to complex. From the usual accounting by the method of addition and subtraction, with their repeated repetition, we came to the concept of multiplication and division. Reducing the repetitive operation of multiplication has become the concept of exponentiation. The first tables of the dependence of numbers on the base and the number of raising to a power were compiled back in the 8th century by the Indian mathematician Varasen. From them, you can count the time of occurrence of logarithms.

Historical sketch

The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation related to multiplication and division of multidigit numbers. 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 primes, 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 to calculate 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. It took a long time before the new operation in algebra acquired its finished form. The definition of the logarithm was given 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 working for the 13th century.

Today we call the base a logarithm of the number x, which is a power of a, to make the number b. This is written in the form of a formula: x = log a (b).

For example, log 3 (9) will be 2. This is obvious if you follow the definition. If 3 is raised to the power of 2, then we get 9.

So, the formulated definition sets 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. Option a = 1 is borderline and is of no interest. Note: 1 is equal to 1 to any degree.

Real value of the logarithm defined only when radix and argument are greater than 0, and the radix must not be equal to 1.

A special place in the field of mathematics play logarithms, which will be named depending on the magnitude of their base:

Rules and restrictions

The fundamental property of logarithms is the rule: the logarithm of the product is equal to the logarithmic sum. log abp = log a (b) + log a (p).

As a variant of this statement will be: log c (b / p) = log c (b) - log c (p), the quotient function is equal to the difference of the functions.

From the previous two rules it is easy to see that: log a (b p) = p * log a (b).

Other properties include:

Comment. Don't 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 has been a rather laborious task. Mathematicians used known formula logarithmic polynomial decomposition theory:

ln (1 + x) = x - (x ^ 2) / 2 + (x ^ 3) / 3 - (x ^ 4) / 4 +… + ((-1) ^ (n + 1)) * (( x ^ n) / n), where n - 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 time consuming and when solving practical problems difficult to implement, then we used pre-compiled tables of logarithms, which greatly accelerated the whole work.

In some cases, specially compiled graphs of logarithms were used, which gave lower accuracy, but significantly accelerated the search. desired value... The curve of the function y = log a (x), built over several points, allows using a regular 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 XIX century acquired a finished look. The most successful device was named slide rule... With all the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and it is difficult to overestimate this. Nowadays, few people are already familiar with this device.

The advent of calculators and computers made it meaningless to use any other device.

Equations and inequalities

For solutions different equations and inequalities using logarithms, the following formulas apply:

The transition from one base to another: log a (b) = log c (b) / log c (a);
Consequently 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 when the base and the argument are both greater 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 inequality sign is preserved; otherwise, it changes.

Examples of tasks

Let's consider several options for using logarithms and their properties. Examples with solving equations:

Consider the option of placing the logarithm in power:

Problem 3. Calculate 25 ^ log 5 (3). Solution: under the conditions of the problem, the record 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 an argument to a function 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 became very important for describing objects the real world... It is difficult to find a science where it is not applied. This fully applies not only to natural, but also humanitarian fields of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and Physics

Historically, mechanics and physics have always evolved using mathematical methods research and at the same time served as a stimulus for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. We will 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 is the final mass.

Another important example- this is the use in the formula of another great scientist Max Planck, which serves to assess the equilibrium state in thermodynamics.

S = k * ln (Ω), where

S - thermodynamic property.
k is Boltzmann's 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. We will also give only two examples:

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 are also not complete without our function.

Psychology and biology

And it’s completely incomprehensible what 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 value of the intensity of the stimulus to the lower value of the intensity.

After the above examples, it is no longer surprising that the topic of logarithms is widely used in biology. 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 rules all the laws. Especially when the laws of nature are associated 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 can be endless. Having mastered the basic laws of this function, you can plunge into the world of infinite wisdom.

We are all familiar with equations with primary grades... There we also learned to solve the simplest examples, and we must admit that they find their application even in higher mathematics... With equations, everything is simple, including square ones. If you have problems with this theme, we strongly recommend that you repeat it.

You probably already passed the logarithms. Nevertheless, we consider it important to tell what it is for those who do not know yet. The logarithm is equated to the degree to which the base must be raised to get the number to the right of the logarithm sign. Let's give an example, based on which, everything will become clear to you.

If you raise 3 to the fourth power, you get 81. Now substitute the numbers by analogy, and you will finally understand how the logarithms are solved. Now it remains only to combine the two considered concepts. Initially, the situation seems extremely difficult, but upon closer examination, the weight falls into place. We are sure that after this short article you will not have any problems in this part of the exam.

Today, there are many ways to solve such structures. We will tell you about the simplest, most effective and most applicable USE assignments. The solution of logarithmic equations should start from the very simple example... The simplest logarithmic equations consist of a function and one variable in it.

It is important to note that x is inside the argument. A and b must be numbers. In this case, you can simply express the function in terms of a number to a power. It looks like this.

Of course, solving the logarithmic equation in this way will lead you to the correct answer. The problem of the vast majority of students in this case is that they do not understand what and where it comes from. As a result, you have to put up with mistakes and not get the desired points. The most offensive mistake will be if you mix up the letters in places. To solve the equation in this way, you need to memorize this standard school formula, because it is difficult to understand it.

To make it easier, you can resort to another method - the canonical form. The idea is very simple. Pay attention to the problem again. Remember that the letter a is a number, not a function or variable. A is not equal to one or greater than zero. There are no restrictions on b. Now we remember one of all the formulas. B can be expressed as follows.

It follows from this that all the original equations with logarithms can be represented as:

We can now drop the logarithms. It will turn out simple construction which we saw earlier.

The convenience of this formula lies in the fact that it can be used in a variety of cases, and not only for the simplest designs.

Don't worry about OOF!

Many experienced mathematicians will notice that we have not paid attention to the domain of definition. The rule is reduced to the fact that F (x) is necessarily greater than 0. No, we did not miss this moment. Now we are talking about another serious advantage of the canonical form.

No unnecessary roots will arise here. If the variable will only appear in one place, then the scope is not necessary. It runs automatically. To verify this statement, consider solving a few simple examples.

How to solve logarithmic equations with different bases

These are already complex logarithmic equations, and the approach to their solution should be special. It rarely turns out to be limited to the notorious canonical form. Let's start our detailed story... We have the following design.

Pay attention to the fraction. It contains the logarithm. If you see this in the assignment, it's worth remembering an interesting trick.

What does it mean? Each logarithm can be represented as a quotient of two logarithms with a convenient base. And this formula has a special case that is applicable with this example (meaning, if c = b).

This is exactly the fraction we see in our example. In this way.

In fact, they turned the fraction over and got a more convenient expression. Remember this algorithm!

Now it is necessary that the logarithmic equation did not contain different bases. Let's imagine the base as a fraction.

In mathematics, there is a rule based on which you can derive a degree from a base. The following construction turns out.

It would seem, what prevents now from turning our expression into a canonical form and solving it in an elementary way? Not so simple. There should be no fractions in front of the logarithm. We fix this situation! The fraction is allowed to be carried out as a degree.

Respectively.

If the bases are the same, we can remove the logarithms and equalize the expressions themselves. So the situation will become much easier than it was. There will remain an elementary equation, which each of us knew how to solve in 8th or even 7th grade. You can make the calculations yourself.

We got the only true root of this logarithmic equation. Examples of solving a logarithmic equation are pretty simple, aren't they? Now you will be able to independently figure out even the most challenging tasks for the preparation and delivery of the exam.

What's the bottom line?

In the case of any logarithmic equations, we proceed from one very important rule... It is necessary to act in such a way as to bring the expression to the maximum simple mind... In this case, you will have more chances not only to solve the task correctly, but also to make it as simple and logical as possible. This is how mathematicians always do.

We strongly discourage you from looking for difficult paths, especially in this case. Remember a few simple rules that will convert any expression. For example, bring two or three logarithms to one base, or derive the degree from the base and win on that.

It is also worth remembering that you need to constantly train in solving logarithmic equations. Gradually, you will move on to more and more complex structures, and this will lead you to confidently solving all variants of problems on the exam. Prepare for your exams well in advance, and good luck!

Decimal logarithm of numbers how to solve. Assignments from the exam

Properties of logarithms

Property 1. Logarithm of the product

Property 2. Logarithm of the quotient

Property 3. Logarithm of the degree

Property 4. Logarithm of the root

The formula for the transition from a logarithm in one base to a logarithm in another base

Comparison of logarithms (inequalities)

How to solve problems with logarithms: examples

What is a logarithm

Logarithms - properties, formulas, how to solve

How to count logarithms

Decimal logarithm

Natural logarithm

Logarithm. Properties of the logarithm (power of the logarithm).

Logarithms. First level.

Logarithms

Decimal and natural logarithms.

Basic properties of logarithms

Basic properties of logarithms

Addition and subtraction of logarithms

Removing the exponent from the logarithm

How to solve logarithms

Moving to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Examples of solving logarithms

Moving to a new foundation

Basic properties of the logarithm

Basic properties of logarithms

Examples for logarithms

Logarithm expressions

Basic properties of logarithms

Addition and subtraction of logarithms

Removing the exponent from the logarithm

Formulas for logarithms. Logarithms are examples of solutions.

Moving to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Basic properties of the logarithm

Common cases of logarithms

Examples for logarithms

Logarithm expressions

Finding the values ​​of logarithms

Logarithms. First level.

Basic properties of logarithms

Addition and subtraction of logarithms

Removing the exponent from the logarithm

How to solve logarithms

Moving to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

What is a logarithm?

If you like this site ...

Historical sketch

Varieties of logarithms

Rules and restrictions

Equations and inequalities

Examples of tasks

Practical use

Logarithmic dependencies

Mechanics and Physics

Chemistry

Psychology and biology

Other areas

Don't worry about OOF!

How to solve logarithmic equations with different bases

What's the bottom line?

Finding the values of logarithms