Difference of logarithms with the same base. Logarithmic Expressions

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.

• Main 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.

Note: decided to get an education of another class study abroad as an option.

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 base 10 logarithm of b can be written as log(b), and the logarithm to the base e (natural logarithm) - 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 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

Degree logarithm is equal to the product of the degree and the logarithm:

If the base of the logarithm is in the exponent, then another formula applies:

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 power of 1/n:

The formula for going from a logarithm in one base to a logarithm in another base

This formula is also often used when solving various tasks for logarithms:

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 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

Tasks with logarithms 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. You can find all examples by searching the site.

What is a logarithm

Logarithms have always been considered a difficult topic in school course mathematics. There are many different definitions of the logarithm, but for some reason most textbooks use the most complex and unfortunate of them.

We will define the logarithm simply and clearly. Let's create a table for this:

So, we have powers of two.

Logarithms - properties, formulas, how to solve

If you take the number from the bottom line, then you can easily find the power to which you have to raise a 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 - in fact, the definition of the logarithm:

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

Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, since 2 6 = 64.

The operation of finding the logarithm of a number to a given base is called. So let's add a new row 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 considered 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 больше степень двойки, тем больше получится число.

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 like this: 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 people confuse where the base is and where the argument is. To avoid unfortunate misunderstandings just take a look at the picture:

Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a 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 there is no confusion.

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:

1. The argument and base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced.
2. The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!

Such restrictions are called valid range(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 are no restrictions on the number b (the value of the logarithm) is not imposed. 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 it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions.

Now consider general scheme logarithm calculations. It consists of three steps:

1. Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
2. Solve the equation for the variable b: x = a b ;
3. 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 that the base be more than one, is very relevant: it reduces the probability of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.

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

A task. Calculate the logarithm: log 5 25

1. Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;

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

2. Received an answer: 2.

A task. Calculate the logarithm:

A task. Calculate the logarithm: log 4 64

1. Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
2. Let's make and solve the equation:
   log 4 64 = b ⇒(2 2) b = 2 6 ⇒2 2b = 2 6 ⇒2b = 6 ⇒ b = 3;
3. Received an answer: 3.

A task. Calculate the logarithm: log 16 1

1. Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
2. Let's make and solve the equation:
   log 16 1 = b ⇒(2 4) b = 2 0 ⇒2 4b = 2 0 ⇒4b = 0 ⇒ b = 0;
3. Received a response: 0.

A task. Calculate the logarithm: log 7 14

1. 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, because 7 1< 14 < 7 2 ;
2. It follows from the previous paragraph that the logarithm is not considered;
3. The answer is no change: log 7 14.

A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If there are at least two distinct factors in the expansion, the number is not an exact power.

A task. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen.

8 \u003d 2 2 2 \u003d 2 3 - the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;

Note also that the prime numbers themselves are always exact powers of themselves.

Decimal logarithm

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

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

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

From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that 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 sense, it is even more important than decimal. This is the natural logarithm.

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

Many will ask: what is the number e? This is an irrational number exact value impossible to find and record. Here are just the first numbers:
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; log 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 irrational. Except, of course, unity: ln 1 = 0.

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

See also:

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

How to represent a number as a logarithm?

We use the definition of a logarithm.

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

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

In the form of a logarithm, you can represent absolutely any number - positive, negative, integer, fractional, rational, irrational:

In order not to confuse a and c in stressful conditions of a test or exam, you can use the following rule to remember:

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

For example, you 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 should be written down, in the base of the degree, and which - up, in the exponent.

The base 3 in the record of the logarithm is at the bottom, which means that when we represent the deuce as a logarithm to the base of 3, we will also write 3 down to the base.

2 is higher than 3. And in the notation of the degree, we write the two 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 the exponent to which the number must be raised. a, To obtain b.

Definition of logarithm can be briefly written like this:

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

Properties of logarithms:

The logarithm of the product:

Logarithm of the quotient from division:

Replacing the base of the logarithm:

Degree logarithm:

root logarithm:

Logarithm with power base:

Decimal and natural logarithms.

Decimal logarithm numbers call the base 10 logarithm of that number and write   lg b
natural logarithm numbers call the logarithm of this number to the base e, where e is an irrational number, approximately equal to 2.7. At the same time, they write ln b.

Other Notes on Algebra and Geometry

Basic properties of logarithms

Basic properties of logarithms

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

These rules must be known - 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 base: log a x and log a y. Then they can be added and subtracted, and:

1. log a x + log a y = log a (x y);
2. 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. Note: key moment here - same grounds. If the bases are different, these rules do not work!

These formulas will help calculate the logarithmic expression even when its individual parts are not considered (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 logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.

A 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.

A 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 made up of "bad" logarithms, which are not considered separately. But after transformations quite normal numbers turn out. Based on this fact, many test papers. Yes, control - similar expressions in all seriousness (sometimes - with virtually no changes) are offered at the exam.

Removing the exponent from the logarithm

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

It is easy to see that the last rule follows their first two. But it's better to remember it anyway - in some cases it will significantly reduce the amount of calculations.

Of course, all these rules make sense if the ODZ 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 before the sign of the logarithm into the logarithm itself.

How to solve logarithms

This is what is most often required.

A task. Find the value of the expression: log 7 49 6 .

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

A task. Find the value of the expression:

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have:

I think the last example needs clarification. Where have logarithms gone? Until the very last moment, we work only with the denominator. They presented the base and the argument of the logarithm standing there in the form of degrees and took out the indicators - they got a “three-story” fraction.

Now let's look at the main fraction. The numerator and denominator have the same number: log 2 7. Since log 2 7 ≠ 0, we can reduce 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 is the answer: 2.

Transition to a new foundation

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

Formulas for transition to a new base come to the rescue. We 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 equality is true:

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

It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.

These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving logarithmic equations and inequalities.

However, there are tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these:

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

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

Now let's flip the second logarithm:

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

A task. Find the value of the expression: log 9 100 lg 3.

The base and argument of the first logarithm are exact powers. Let's write it down and get rid of the indicators:

Now let's get rid of decimal logarithm, moving to a 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's just the value of the logarithm.

The second formula is actually a paraphrased definition. It's called like this:

Indeed, what will happen if the number b is raised to such a degree that the number b in this degree gives the number a? That's right: this is the same number a. Read this paragraph carefully again - many people “hang” on it.

Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution.

A task. Find the value of the expression:

Note that log 25 64 = log 5 8 - just took out the square from the base and the argument of the logarithm. Given the rules for multiplying powers with the same base, we get:

If someone is not in the know, this was a real task from the Unified State Examination 🙂

Logarithmic unit and logarithmic zero

In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm. They are constantly found in problems and, surprisingly, create problems even for "advanced" students.

1. log a a = 1 is. Remember once and for all: the logarithm to any base a from that base itself is equal to one.
2. log a 1 = 0 is. The base a can be anything, but if the argument is 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.

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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 related to multiplication and division 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 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 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. Engineers long time for these purposes, the so-called graph paper was used.

In the 17th century, the first auxiliary analog computing conditions appeared, which to XIX century acquired a finished look. The most successful device was called the slide rule. Despite the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and this is difficult to overestimate. Currently, few people are familiar with this device.

The advent of calculators and computers made it pointless to use any other devices.

Equations and inequalities

The following formulas are used to solve various equations and inequalities using logarithms:

• Transition from one base to another: log a(b) = log c(b) / log c(a);
• As a consequence of the previous version: log a(b) = 1 / log b(a).

To solve inequalities, it is useful to know:

• The value of the logarithm will 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 from real life that the logarithm suddenly acquired great importance to describe objects real world. It is difficult to find a science where it is not used. This fully applies not only to 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 methods research and at the same time served as an incentive for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. Here are just two examples of descriptions 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.

\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)

Let's explain it easier. For example, \(\log_(2)(8)\) is equal to the power \(2\) must be raised to to get \(8\). From this it is clear that \(\log_(2)(8)=3\).

Examples:		\(\log_(5)(25)=2\)		because \(5^(2)=25\)
		\(\log_(3)(81)=4\)		because \(3^(4)=81\)
		\(\log_(2)\)\(\frac(1)(32)\) \(=-5\)		because \(2^(-5)=\)\(\frac(1)(32)\)

Argument and base of the logarithm

Any logarithm has the following "anatomy":

The argument of the logarithm is usually written at its level, and the base is written in subscript closer to the sign of the logarithm. And this entry is read like this: "the logarithm of twenty-five to the base of five."

How to calculate the logarithm?

To calculate the logarithm, you need to answer the question: to what degree should the base be raised to get the argument?

For example, calculate the logarithm: a) \(\log_(4)(16)\) b) \(\log_(3)\)\(\frac(1)(3)\) c) \(\log_(\sqrt (5))(1)\) d) \(\log_(\sqrt(7))(\sqrt(7))\) e) \(\log_(3)(\sqrt(3))\)

a) To what power must \(4\) be raised to get \(16\)? Obviously the second. That's why:

\(\log_(4)(16)=2\)

\(\log_(3)\)\(\frac(1)(3)\) \(=-1\)

c) To what power must \(\sqrt(5)\) be raised to get \(1\)? And what degree makes any number a unit? Zero, of course!

\(\log_(\sqrt(5))(1)=0\)

d) To what power must \(\sqrt(7)\) be raised to get \(\sqrt(7)\)? In the first - any number in the first degree is equal to itself.

\(\log_(\sqrt(7))(\sqrt(7))=1\)

e) To what power must \(3\) be raised to get \(\sqrt(3)\)? From we know that is a fractional power, which means Square root is the degree \(\frac(1)(2)\) .

\(\log_(3)(\sqrt(3))=\)\(\frac(1)(2)\)

Example : Calculate the logarithm \(\log_(4\sqrt(2))(8)\)

Solution :

\(\log_(4\sqrt(2))(8)=x\)		We need to find the value of the logarithm, let's denote it as x. Now let's use the definition of the logarithm: \(\log_(a)(c)=b\) \(\Leftrightarrow\) \(a^(b)=c\)
\((4\sqrt(2))^(x)=8\)		What links \(4\sqrt(2)\) and \(8\)? Two, because both numbers can be represented by twos: \(4=2^(2)\) \(\sqrt(2)=2^(\frac(1)(2))\) \(8=2^(3)\)
\(((2^(2)\cdot2^(\frac(1)(2))))^(x)=2^(3)\)		On the left, we use the degree properties: \(a^(m)\cdot a^(n)=a^(m+n)\) and \((a^(m))^(n)=a^(m\cdot n)\)
\(2^(\frac(5)(2)x)=2^(3)\)		The bases are equal, we proceed to the equality of indicators
\(\frac(5x)(2)\) \(=3\)		Multiply both sides of the equation by \(\frac(2)(5)\)
		The resulting root is the value of the logarithm

Answer : \(\log_(4\sqrt(2))(8)=1,2\)

Why was the logarithm invented?

To understand this, let's solve the equation: \(3^(x)=9\). Just match \(x\) to make the equality work. Of course, \(x=2\).

Now solve the equation: \(3^(x)=8\). What is x equal to? That's the point.

The most ingenious will say: "X is a little less than two." How exactly is this number to be written? To answer this question, they came up with the logarithm. Thanks to him, the answer here can be written as \(x=\log_(3)(8)\).

I want to emphasize that \(\log_(3)(8)\), as well as any logarithm is just a number. Yes, it looks unusual, but it is short. Because if we wanted to write it in the form decimal fraction, then it would look like this: \(1.892789260714.....\)

Example : Solve the equation \(4^(5x-4)=10\)

Solution :

\(4^(5x-4)=10\)		\(4^(5x-4)\) and \(10\) cannot be reduced to the same base. So here you can not do without the logarithm. Let's use the definition of the logarithm: \(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
\(\log_(4)(10)=5x-4\)		Flip the equation so x is on the left
\(5x-4=\log_(4)(10)\)		Before us. Move \(4\) to the right. And don't be afraid of the logarithm, treat it like a regular number.
\(5x=\log_(4)(10)+4\)		Divide the equation by 5
\(x=\)\(\frac(\log_(4)(10)+4)(5)\)		Here is our root. Yes, it looks unusual, but the answer is not chosen.

Answer : \(\frac(\log_(4)(10)+4)(5)\)

Decimal and natural logarithms

As stated in the definition of the logarithm, its base can be any positive number except one \((a>0, a\neq1)\). And among all the possible bases, there are two that occur so often that a special short notation was invented for logarithms with them:

Natural logarithm: a logarithm whose base is the Euler number \(e\) (equal to approximately \(2.7182818…\)), and the logarithm is written as \(\ln(a)\).

That is, \(\ln(a)\) is the same as \(\log_(e)(a)\)

Decimal logarithm: A logarithm whose base is 10 is written \(\lg(a)\).

That is, \(\lg(a)\) is the same as \(\log_(10)(a)\), where \(a\) is some number.

Basic logarithmic identity

Logarithms have many properties. One of them is called "Basic logarithmic identity" and looks like this:

\(a^(\log_(a)(c))=c\)

This property follows directly from the definition. Let's see how exactly this formula appeared.

Recall the short definition of the logarithm:

if \(a^(b)=c\), then \(\log_(a)(c)=b\)

That is, \(b\) is the same as \(\log_(a)(c)\). Then we can write \(\log_(a)(c)\) instead of \(b\) in the formula \(a^(b)=c\) . It turned out \(a^(\log_(a)(c))=c\) - the main logarithmic identity.

You can find the rest of the properties of logarithms. With their help, you can simplify and calculate the values ​​of expressions with logarithms, which are difficult to calculate directly.

Example : Find the value of the expression \(36^(\log_(6)(5))\)

Solution :

Answer : \(25\)

How to write a number as a logarithm?

As mentioned above, any logarithm is just a number. The converse is also true: any number can be written as a logarithm. For example, we know that \(\log_(2)(4)\) is equal to two. Then you can write \(\log_(2)(4)\) instead of two.

But \(\log_(3)(9)\) is also equal to \(2\), so you can also write \(2=\log_(3)(9)\) . Similarly with \(\log_(5)(25)\), and with \(\log_(9)(81)\), etc. That is, it turns out

\(2=\log_(2)(4)=\log_(3)(9)=\log_(4)(16)=\log_(5)(25)=\log_(6)(36)=\ log_(7)(49)...\)

Thus, if we need, we can write the two as a logarithm with any base anywhere (even in an equation, even in an expression, even in an inequality) - we just write the squared base as an argument.

It's the same with a triple - it can be written as \(\log_(2)(8)\), or as \(\log_(3)(27)\), or as \(\log_(4)(64) \) ... Here we write the base in the cube as an argument:

\(3=\log_(2)(8)=\log_(3)(27)=\log_(4)(64)=\log_(5)(125)=\log_(6)(216)=\ log_(7)(343)...\)

And with four:

\(4=\log_(2)(16)=\log_(3)(81)=\log_(4)(256)=\log_(5)(625)=\log_(6)(1296)=\ log_(7)(2401)...\)

And with minus one:

\(-1=\) \(\log_(2)\)\(\frac(1)(2)\) \(=\) \(\log_(3)\)\(\frac(1)( 3)\) \(=\) \(\log_(4)\)\(\frac(1)(4)\) \(=\) \(\log_(5)\)\(\frac(1 )(5)\) \(=\) \(\log_(6)\)\(\frac(1)(6)\) \(=\) \(\log_(7)\)\(\frac (1)(7)\)\(...\)

And with one third:

\(\frac(1)(3)\) \(=\log_(2)(\sqrt(2))=\log_(3)(\sqrt(3))=\log_(4)(\sqrt( 4))=\log_(5)(\sqrt(5))=\log_(6)(\sqrt(6))=\log_(7)(\sqrt(7))...\)

Any number \(a\) can be represented as a logarithm with base \(b\): \(a=\log_(b)(b^(a))\)

Example : Find the value of an expression \(\frac(\log_(2)(14))(1+\log_(2)(7))\)

Solution :

Answer : \(1\)