How to swap the base and number of a logarithm. Logarithm

Today we will talk about logarithmic formulas and we will give indicative solution examples.

They themselves imply solution patterns according to the basic properties of logarithms. Before applying logarithm formulas to solve, let us remind you of all the properties:

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

Examples of solving logarithms based on formulas.

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

According to definitions of 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- this is an ordinary logarithm, the base of which is 10. It is denoted as lg.

log 10 100 = 2, because 10 2 = 100

Natural logarithm- also the usual logarithm logarithm, but with the base e (e = 2.71828... - irrational number). Denoted as ln.

It is advisable to memorize 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.

Basics logarithmic identity
a log a b = b
8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9
Logarithm of the product equal to the sum 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 power of a logarithmic number and the base of the logarithm
Exponent of logarithm log numbers 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 formulas for logarithms are not as complicated as they seem. Now, having looked at examples of solving logarithms, we can move on to logarithmic equations. We will look at 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: we decided to get a different class of education and study abroad as an option.

The main properties are given natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, expansion in power series and representation of the function ln x using complex numbers.

Definition

Natural logarithm is the function y = ln x, the inverse of the exponential, x = e y, and is the logarithm to the base of the number e: ln x = log e x.

The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.

Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.

Graph of the function y = ln x.

Graph of natural logarithm (functions y = ln x) is obtained from the exponential graph mirror image relative to the straight line y = x.

The natural logarithm is defined at positive values variable x. It increases monotonically in its domain of definition.

At x → 0 the limit of the natural logarithm is minus infinity (-∞).

As x → + ∞, the limit of the natural logarithm is plus infinity (+ ∞). For large x, the logarithm increases quite slowly. Any power function x a s positive indicator degree a grows faster than the logarithm.

Properties of the natural logarithm

Domain of definition, set of values, extrema, increase, decrease

The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.

ln x values

ln 1 = 0

Basic formulas for natural logarithms

Formulas following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Any logarithm can be expressed in terms of natural logarithms using the base substitution formula:

Proofs of these formulas are presented in the section "Logarithm".

Inverse function

The inverse of the natural logarithm is the exponent.

If , then

If, then.

Derivative ln x

Derivative of the natural logarithm:
.
Derivative of the natural logarithm of modulus x:
.
Derivative of nth order:
.
Deriving formulas > > >

Integral

The integral is calculated by integration by parts:
.
So,

Expressions using complex numbers

Consider the function of the complex variable z:
.
Let's express the complex variable z via module r and argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If you put
, where n is an integer,
it will be the same number for different n.

Therefore, the natural logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Logarithmic expressions, solving examples. In this article we will look at problems related to solving logarithms. The tasks ask the question of finding the meaning of an expression. It should be noted that the concept of logarithm is used in many tasks and understanding its meaning is extremely important. As for the Unified State Examination, the logarithm is used when solving equations, in applied problems, also in tasks related to the study of functions.

Let us give examples to understand the very meaning of the logarithm:

Basic logarithmic identity:

Properties of logarithms that must always be remembered:

*The logarithm of the product is equal to the sum of the logarithms of the factors.

* * *

*The logarithm of a quotient (fraction) is equal to the difference between the logarithms of the factors.

* * *

*Logarithm of degree equal to the product exponent by the logarithm of its base.

* * *

*Transition to a new foundation

* * *

More properties:

* * *

The calculation of logarithms is closely related to the use of properties of exponents.

Let's list some of them:

The essence of this property lies in the fact that when transferring the numerator to the denominator and vice versa, the sign of the exponent changes to the opposite. For example:

A corollary from this property:

* * *

When raising a power to a power, the base remains the same, but the exponents are multiplied.

* * *

As you have seen, the concept of a logarithm itself is simple. The main thing is what is needed good practice, which gives a certain skill. Of course, knowledge of formulas is required. If the skill in converting elementary logarithms has not been developed, then when solving simple tasks you can easily make a mistake.

Practice, solve the simplest examples from the mathematics course first, then move on to more complex ones. In the future, I will definitely show how “scary” logarithms are solved; they won’t appear on the Unified State Examination, but they are of interest, don’t miss them!

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell me about the site on social networks.

As society developed and production became more complex, mathematics also developed. Movement from simple to complex. From conventional accounting by the method of addition and subtraction, with their repeated many times, came to the concept of multiplication and division. Reducing the repeated operation of multiplication 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 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 multi-digit numbers. The ancient tables were of great service. They made it possible to replace complex operations with simpler ones - addition and subtraction. Big step The work of the mathematician Michael Stiefel, published in 1544, took the lead, 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 new term"logarithm of a number." New complex tables for calculating 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 throughout three centuries. A lot of time passed 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 computer, did humanity abandon the ancient tables that had worked successfully throughout the 13th centuries.

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

For example, log 3(9) would 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 sets only one restriction: the numbers a and b must be real.

Types of logarithms

The classic definition is called the real logarithm and is actually the solution to the equation a x = b. Option a = 1 is borderline and is not of interest. Attention: 1 to any power is equal to 1.

Real value of logarithm defined only when the base and the argument are 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 size 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 there 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. There is no need to make a common mistake - the logarithm of a sum is not equal to the sum of logarithms.

For many centuries, the operation of finding a logarithm was a rather time-consuming task. Mathematicians used well-known formula logarithmic theory of polynomial expansion:

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 about the transition from one base to another and the property of the logarithm of the product.

Since this method is very labor-intensive and when deciding practical problems difficult to implement, we used pre-compiled tables of logarithms, which significantly speeded up all the work.

In some cases, specially designed logarithm graphs were used, which gave less accuracy, but significantly speeded up the search desired value. The curve of the function y = log a(x), constructed over several points, allows you to use a regular ruler to find the value of the function at any other point. Engineers long time For these purposes, so-called graph paper was used.

In the 17th century, the first auxiliary analog computing conditions appeared, which 19th 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 the use of any other devices pointless.

Equations and inequalities

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

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

To solve inequalities it is useful to know:

The value of the logarithm will be positive only if the base and argument are both greater or less than one; if at least one condition is violated, the logarithm value will be negative.
If the logarithm function is applied to the right and left sides of an inequality, and the base of the logarithm more than one, then the inequality sign is preserved; V otherwise he is changing.

Sample problems

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

Consider the option of placing the logarithm in a power:

Problem 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the entry 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 equal to 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 natural, but also to humanitarian 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. Let's give just two examples of descriptions physical laws using logarithm.

Solve a calculation problem like this complex size How the speed of a rocket can be determined by applying the Tsiolkovsky formula, which laid the foundation for the theory of space exploration:

V = I * ln (M1/M2), where

V – final speed aircraft.
I – specific impulse of the engine.
M 1 – initial mass of the rocket.
M 2 – final mass.

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

S = k * ln (Ω), where

S – thermodynamic property.
k – Boltzmann constant.
Ω is the statistical weight of different states.

Chemistry

Less obvious is the use of formulas in chemistry containing the ratio of logarithms. Let's give just 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 autolysis index and the acidity of the solution also cannot be done without our function.

Psychology and biology

And it’s not at all clear what psychology has to do with it. It turns out that the strength of sensation is well described by this function as inverse relation stimulus intensity values to the lower intensity value.

After the above examples, it is no longer surprising that the topic of logarithms is widely used in biology. Entire volumes could 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 laws. Especially when the laws of nature are related to geometric progression. It’s worth turning 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 principles of this function, you can plunge into the world of infinite wisdom.

main 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 a logarithm is a power? 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 ODZ of the logarithm is observed: a > 0, a ≠ 1, x >

Task. Find the meaning of the expression:

Transition to a new foundation

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

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 equal to 2.7 and twice the year of birth of Leo Nikolaevich Tolstoy.

Basic properties of logarithms

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

Examples for logarithms

Logarithm expressions

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

Using properties 3.5 we calculate

4. Where .

Example 2. Find x if

Example 3. Let the value of logarithms be given

Calculate log(x) if

Basic properties of logarithms

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

You definitely need to know these rules - without them not a single serious problem can be solved. logarithmic problem. In addition, there are very few of them - you can learn everything in one day. So let's get started.

Adding and subtracting 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).

So, the sum of logarithms is equal to the logarithm of the product, and the difference is equal to the logarithm of the quotient. Note: key moment Here - identical grounds. If the reasons are different, these rules do not work!

These formulas will help you calculate 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 logarithms have the same bases, 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 made up of “bad” logarithms, which are not calculated separately. But after the transformations they turn out quite normal numbers. Many are built on this fact test papers. What about the controls? similar expressions in all seriousness (sometimes with virtually no changes) are offered on the Unified State Examination.

Extracting the exponent from the logarithm

It's easy to notice that last rule follows the 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 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 before the logarithm sign into the logarithm itself. 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 a logarithm, the base and argument of which are exact powers: 16 = 24; 49 = 72. We have:

I think to last example clarification required. Where have logarithms gone? Until the very last moment we work only with the denominator.

Logarithm formulas. Logarithms examples solutions.

We presented the base and argument of the logarithm standing there in the form of powers and took out the exponents - we got a “three-story” fraction.

Now let's look at the main fraction. The numerator and denominator contain the same number: log2 7. Since log2 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 is what was done. The result was 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 reasons are different? What if they are not exact powers of the same number?

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

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

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

From the second formula it follows that the base and argument of the logarithm can be swapped, but in this case the entire expression is “turned over”, i.e. the logarithm appears in the denominator.

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

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

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

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

Now let’s “reverse” the second logarithm:

Since the product does not change when rearranging factors, we calmly multiplied four and two, and then dealt with logarithms.

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

The base and argument of the first logarithm are exact powers. Let's write this 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 solution process it is necessary to represent a number as a logarithm to a given base. In this case, the following 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 a logarithm value.

The second formula is actually a paraphrased definition. That's what it's called: .

In fact, 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: the result is the same number a. Read this paragraph carefully again - many people get stuck on it.

Like formulas for moving 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 - simply took the square from the base and argument of the logarithm. Considering the rules for multiplying powers with the same basis, we get:

If anyone doesn’t know, this was a real task from the Unified State 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 constantly appear in problems and, surprisingly, create problems even for “advanced” students.

logaa = 1 is. Remember once and for all: the logarithm to any base a of that base itself is equal to one.
loga 1 = 0 is. The base a can be anything, but if the argument contains one - logarithm equal to 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

It is necessary to know the above properties, since almost all problems and examples related to logarithms are solved on their basis. The rest of the exotic properties can be derived through mathematical manipulations with these formulas

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

When calculating the formula for the sum and difference of logarithms (3.4) you come across 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 logarithm to base ten is usually called the decimal logarithm and is simply denoted by lg(x).

It is clear from the recording that the basics are not written in the recording. For example

A natural logarithm is a logarithm whose base is an exponent (denoted by ln(x)).

The exponent is 2.718281828…. To remember the exponent, you can study the rule: the exponent is equal to 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 logarithm to base two is denoted by

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

Integral or antiderivative logarithm determined by dependency

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

Examples for logarithms

Logarithm expressions

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

Using properties 3.5 we calculate

2.
By the property of difference of logarithms we have

3.
Using properties 3.5 we find

4. Where .

By the look complex expression using a number of rules is simplified to form

Finding logarithm values

Example 2. Find x if

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

We put it on record and mourn

Since the bases are equal, we equate the expressions

Logarithms. First level.

Let the value of logarithms be given

Calculate log(x) if

Solution: Let's take a logarithm of the variable to write the logarithm through the sum of its terms

This is just the beginning of our acquaintance with logarithms and their properties. Practice calculations, enrich your practical skills - you will soon need the knowledge you gain 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

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 main properties.

You definitely need to know these rules - without them, not a single serious logarithmic problem can be solved. In addition, there are very few of them - you can learn everything in one day. So let's get started.

Adding and subtracting 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).

So, the sum of logarithms is equal to the logarithm of the product, and the difference is equal to 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 considered (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 logarithms have the same bases, 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 made up of “bad” logarithms, which are not calculated separately. But after the transformations, completely normal numbers are obtained. Many tests are based on this fact. Yes, test-like expressions are offered in all seriousness (sometimes with virtually no changes) on the Unified State Examination.

Extracting the exponent from the logarithm

Now let's complicate the task a little. What if the base or argument of a logarithm is a power? 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 the first two. But it’s better to remember it anyway - in some cases it will significantly reduce the amount of calculations.

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

I think the last example requires some clarification. Where have logarithms gone? Until the very last moment we work only with the denominator. We presented the base and argument of the logarithm standing there in the form of powers and took out the exponents - we got a “three-story” fraction.

Transition to a new foundation

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

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

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

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 problems that cannot be solved at all except by moving to a new foundation. Let's look at a couple of these:

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

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

Now let’s “reverse” the second logarithm:

Since the product does not change when rearranging factors, we calmly multiplied four and two, and then dealt with logarithms.

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

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

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

Basic logarithmic identity

Often in the solution process it is necessary to represent a number as a logarithm to a given base. In this case, the following 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 a logarithm value.

The second formula is actually a paraphrased definition. That's what it's called: .

Like formulas for moving 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 - simply took the square from the base and argument of the logarithm. Taking into account the rules for multiplying powers with the same base, we get:

If anyone doesn’t know, this was a real task from the Unified State Exam :)

Logarithmic unit and logarithmic zero

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