If the bases of the logarithms are the same, then the exponents. Always be in the mood

The logarithm of the number b to the base a is the exponent to which you need to raise the number a to get the number b.

If , then .

The logarithm is extremely important mathematical quantity, since the logarithmic calculus allows not only solving exponential equations, but also operating with exponents, differentiating exponential and logarithmic functions, integrating them and bringing them to a more acceptable form to be calculated.

In contact with

All properties of logarithms are directly related to the properties of exponential functions. For example, the fact that means that:

It should be noted that when solving specific problems, the properties of logarithms may be more important and useful than the rules for working with powers.

Here are some identities:

Here are the main algebraic expressions:

;

Attention! can only exist for x>0, x≠1, y>0.

Let's try to understand the question of what natural logarithms are. Separate interest in mathematics represent two types- the first has the number "10" at the base, and is called the "decimal logarithm". The second is called natural. The base of the natural logarithm is the number e. It is about him that we will talk in detail in this article.

Designations:

lg x - decimal;
ln x - natural.

Using the identity, we can see that ln e = 1, as well as that lg 10=1.

natural log graph

We construct a graph of the natural logarithm in the standard classical way by points. If you wish, you can check whether we are building a function correctly by examining the function. However, it makes sense to learn how to build it "manually" in order to know how to correctly calculate the logarithm.

Function: y = log x. Let's write a table of points through which the graph will pass:

Let us explain why we chose such values of the argument x. It's all about identity: For a natural logarithm, this identity will look like this:

For convenience, we can take five reference points:

;

Thus, counting natural logarithms is a fairly simple task, moreover, it simplifies the calculation of operations with powers, turning them into normal multiplication.

Having built a graph by points, we get an approximate graph:

The domain of the natural logarithm (that is, all valid values of the X argument) is all numbers greater than zero.

Attention! The domain of the natural logarithm includes only positive numbers! The scope does not include x=0. This is impossible based on the conditions for the existence of the logarithm.

The range of values (i.e. all valid values of the function y = ln x) is all numbers in the interval .

natural log limit

Studying the graph, the question arises - how does the function behave when y<0.

Obviously, the graph of the function tends to cross the y-axis, but will not be able to do this, since the natural logarithm of x<0 не существует.

Natural limit log can be written like this:

Formula for changing the base of a logarithm

Dealing with a natural logarithm is much easier than dealing with a logarithm that has an arbitrary base. That is why we will try to learn how to reduce any logarithm to a natural one, or express it in an arbitrary base through natural logarithms.

Let's start with the logarithmic identity:

Then any number or variable y can be represented as:

where x is any number (positive according to the properties of the logarithm).

This expression can be logarithmized on both sides. Let's do this with an arbitrary base z:

Let's use the property (only instead of "with" we have an expression):

From here we get the universal formula:

In particular, if z=e, then:

We managed to represent the logarithm to an arbitrary base through the ratio of two natural logarithms.

We solve problems

In order to better navigate in natural logarithms, consider examples of several problems.

Task 1. It is necessary to solve the equation ln x = 3.

Decision: Using the definition of the logarithm: if , then , we get:

Task 2. Solve the equation (5 + 3 * ln (x - 3)) = 3.

Solution: Using the definition of the logarithm: if , then , we get:

Once again, we apply the definition of the logarithm:

Thus:

You can calculate the answer approximately, or you can leave it in this form.

Task 3. Solve the equation.

Decision: Let's make a substitution: t = ln x. Then the equation will take the following form:

We have a quadratic equation. Let's find its discriminant:

First root of the equation:

Second root of the equation:

Remembering that we made the substitution t = ln x, we get:

In statistics and probability theory, logarithmic quantities are very common. This is not surprising, because the number e - often reflects the growth rate of exponential values.

In computer science, programming and computer theory, logarithms are quite common, for example, in order to store N bits in memory.

In the theories of fractals and dimensions, logarithms are constantly used, since the dimensions of fractals are determined only with their help.

In mechanics and physics there is no section where logarithms were not used. The barometric distribution, all the principles of statistical thermodynamics, the Tsiolkovsky equation and so on are processes that can only be mathematically described using logarithms.

In chemistry, the logarithm is used in the Nernst equations, descriptions of redox processes.

Amazingly, even in music, in order to find out the number of parts of an octave, logarithms are used.

Natural logarithm Function y=ln x its properties

Proof of the main property of the natural logarithm

basic properties.

logax + logay = log(x y);
logax − logay = log(x: y).

same grounds

log6 4 + log6 9.

Now let's complicate the task a little.

Examples of solving logarithms

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:

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

Task. Find the value 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 value 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 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

Take the logarithm of 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 logarithms be given

Calculate log(x) if

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: logax and logay. Then they can be added and subtracted, and:

logax + logay = log(x y);
logax − logay = log(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 - 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:

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

Removing the exponent from the logarithm

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. 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 according to the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the value of the expression:

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 24; 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.

Formulas of logarithms. Logarithms are examples of solutions.

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: 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 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 logax 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:

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

Note that the arguments of both logarithms are exact exponents. 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 permutation of factors, we calmly multiplied four and two, and then figured out the 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 it 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 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.

Task. Find the value of the expression:

Note that log25 64 = log5 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.

logaa = 1 is. Remember once and for all: the logarithm to any base a from that base itself 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 solved based on logarithms. The remaining exotic properties can be derived 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 deuce.
The base ten logarithm is usually called the base ten logarithm and is simply denoted lg(x).

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

The natural logarithm is the logarithm whose basis is the exponent (denoted 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 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 logarithm is determined by the dependence

The above 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 the school curriculum and universities.

Examples for logarithms

Take the logarithm of expressions

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

By properties 3,5 we calculate

2.
By the difference property of logarithms, we have

3.
Using properties 3.5 we find

4. where .

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

Finding Logarithm Values

Example 2 Find x if

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

Substitute in the record and mourn

Since the bases are equal, we equate the expressions

Logarithms. First level.

Let the value of the logarithms be given

Calculate log(x) if

Solution: Take the logarithm of the variable to write the logarithm through the sum of the terms

This is just the beginning of acquaintance with logarithms and their properties. Practice calculations, enrich your practical skills - you will soon need the acquired knowledge to solve logarithmic equations. Having studied the basic methods for solving such equations, we will expand your knowledge for another equally important topic - logarithmic inequalities ...

Basic properties of logarithms

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: logax and logay. Then they can be added and subtracted, and:

logax + logay = log(x y);
logax − logay = log(x: y).

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:

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

Since the bases of 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.

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.

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 according to the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the value of the expression:

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 24; 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.

Transition to a new foundation

Formulas for transition to a new base come to the rescue. We 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 put 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 tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these:

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

Note that the arguments of both logarithms are exact exponents. 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 permutation of factors, we calmly multiplied four and two, and then figured out the 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 it 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 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:

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

Task. Find the value of the expression:

Note that log25 64 = log5 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

logaa = 1 is. Remember once and for all: the logarithm to any base a from that base itself 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.

The section of logarithms is of great importance in the school course "Mathematical Analysis". Tasks for logarithmic functions are based on other principles than tasks for inequalities and equations. Knowledge of the definitions and basic properties of the concepts of logarithm and logarithmic function will ensure the successful solution of typical USE problems.

Before proceeding to explain what a logarithmic function is, it is worth referring to the definition of a logarithm.

Let's look at a specific example: a log a x = x, where a › 0, a ≠ 1.

The main properties of logarithms can be listed in several points:

Logarithm

Logarithm is a mathematical operation that allows using the properties of a concept to find the logarithm of a number or expression.

Examples:

Logarithm function and its properties

The logarithmic function has the form

Immediately, we note that the function graph can be increasing for a › 1 and decreasing for 0 ‹ a ‹ 1. Depending on this, the function curve will have one form or another.

Here are the properties and method for plotting graphs of logarithms:

the domain of f(x) is the set of all positive numbers, i.e. x can take any value from the interval (0; + ∞);
ODZ functions - the set of all real numbers, i.e. y can be equal to any number from the interval (- ∞; +∞);
if the base of the logarithm a > 1, then f(x) increases over the entire domain of definition;
if the base of the logarithm is 0 ‹ a ‹ 1, then F is decreasing;
the logarithmic function is neither even nor odd;
the graph curve always passes through the point with coordinates (1;0).

Building both types of graphs is very simple, let's look at the process using an example

First you need to remember the properties of a simple logarithm and its function. With their help, you need to build a table for specific x and y values. Then, on the coordinate axis, the obtained points should be marked and connected by a smooth line. This curve will be the required graph.

The logarithmic function is the inverse of the exponential function given by y= a x . To verify this, it is enough to draw both curves on the same coordinate axis.

Obviously, both lines are mirror images of each other. By constructing a straight line y = x, you can see the axis of symmetry.

In order to quickly find the answer to the problem, you need to calculate the values of the points for y = log 2⁡ x, and then simply move the origin of the coordinate points three divisions down the OY axis and 2 divisions to the left along the OX axis.

As proof, we will build a calculation table for the points of the graph y = log 2 ⁡ (x + 2) -3 and compare the obtained values with the figure.

As you can see, the coordinates from the table and the points on the graph match, therefore, the transfer along the axes was carried out correctly.

Examples of solving typical USE problems

Most of the test tasks can be divided into two parts: finding the domain of definition, specifying the type of function according to the graph drawing, determining whether the function is increasing / decreasing.

For a quick answer to tasks, it is necessary to clearly understand that f(x) increases if the exponent of the logarithm a > 1, and decreases - when 0 ‹ a ‹ 1. However, not only the base, but also the argument can greatly affect the form of the function curve.

F(x) marked with a check mark are the correct answers. Doubts in this case are caused by examples 2 and 3. The “-” sign in front of log changes increasing to decreasing and vice versa.

Therefore, the graph y=-log 3⁡ x decreases over the entire domain of definition, and y= -log (1/3) ⁡x increases, despite the fact that the base is 0 ‹ a ‹ 1.

Answer: 3,4,5.

Answer: 4.

These types of tasks are considered easy and are estimated at 1-2 points.

Task 3.

Determine whether the function is decreasing or increasing and indicate the scope of its definition.

Y = log 0.7 ⁡(0.1x-5)

Since the base of the logarithm is less than one but greater than zero, the function of x is decreasing. According to the properties of the logarithm, the argument must also be greater than zero. Let's solve the inequality:

Answer: the domain of definition D(x) is the interval (50; + ∞).

Answer: 3, 1, OX axis, to the right.

Such tasks are classified as average and are estimated at 3-4 points.

Task 5. Find the range for a function:

It is known from the properties of the logarithm that the argument can only be positive. Therefore, we calculate the area of admissible values of the function. To do this, it will be necessary to solve a system of two inequalities.

Logarithmic expressions, solution of examples. In this article, we will consider problems related to solving logarithms. The tasks raise the question of finding the value of the expression. It should be noted that the concept of the logarithm is used in many tasks and it is extremely important to understand its meaning. As for the USE, the logarithm is used in solving equations, in applied problems, and also in tasks related to the study of functions.

Here are examples to understand the very meaning of the logarithm:

Basic logarithmic identity:

Properties of logarithms that you must always remember:

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

* * *

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

* * *

* The logarithm of the degree is equal to the product of the exponent and the logarithm of its base.

* * *

*Transition to new base

* * *

More properties:

* * *

Computing logarithms is closely related to using the properties of exponents.

We list some of them:

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

Consequence of this property:

* * *

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

* * *

As you can see, the very concept of the logarithm is simple. The main thing is that good practice is needed, which gives a certain skill. Certainly knowledge of formulas is obligatory. If the skill in converting elementary logarithms is not formed, then when solving simple tasks, one can easily make a mistake.

Practice, solve the simplest examples from the math course first, then move on to more complex ones. In the future, I will definitely show how the “ugly” logarithms are solved, there will be no such ones at the exam, but they are of interest, do not miss it!

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh

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

The logarithm of a positive number b to base a (a>0, a is not equal to 1) is a number c such that a c = b: log a b = c ⇔ a c = b (a > 0, a ≠ 1, b > 0)

Note that the logarithm of a non-positive number is not defined. Also, the base of the logarithm must be a positive number, not equal to 1. For example, if we square -2, we get the number 4, but this does not mean that the base -2 logarithm of 4 is 2.

Basic logarithmic identity

a log a b = b (a > 0, a ≠ 1) (2)

It is important that the domains of definition of the right and left parts of this formula are different. The left side is defined only for b>0, a>0 and a ≠ 1. The right side is defined for any b, and does not depend on a at all. Thus, the application of the basic logarithmic "identity" in solving equations and inequalities can lead to a change in the DPV.

Two obvious consequences of the definition of the logarithm

log a a = 1 (a > 0, a ≠ 1) (3)
log a 1 = 0 (a > 0, a ≠ 1) (4)

Indeed, when raising the number a to the first power, we get the same number, and when raising it to the zero power, we get one.

The logarithm of the product and the logarithm of the quotient

log a (b c) = log a b + log a c (a > 0, a ≠ 1, b > 0, c > 0) (5)

Log a b c = log a b − log a c (a > 0, a ≠ 1, b > 0, c > 0) (6)

I would like to warn schoolchildren against the thoughtless use of these formulas when solving logarithmic equations and inequalities. When they are used "from left to right", the ODZ narrows, and when moving from the sum or difference of logarithms to the logarithm of the product or quotient, the ODZ expands.

Indeed, the expression log a (f (x) g (x)) is defined in two cases: when both functions are strictly positive or when f(x) and g(x) are both less than zero.

Transforming this expression into the sum log a f (x) + log a g (x) , we are forced to restrict ourselves only to the case when f(x)>0 and g(x)>0. There is a narrowing of the range of admissible values, and this is categorically unacceptable, since it can lead to the loss of solutions. A similar problem exists for formula (6).

The degree can be taken out of the sign of the logarithm

log a b p = p log a b (a > 0, a ≠ 1, b > 0) (7)

And again I would like to call for accuracy. Consider the following example:

Log a (f (x) 2 = 2 log a f (x)

The left side of the equality is obviously defined for all values of f(x) except zero. The right side is only for f(x)>0! Taking the power out of the logarithm, we again narrow the ODZ. The reverse procedure leads to an expansion of the range of admissible values. All these remarks apply not only to the power of 2, but also to any even power.

Formula for moving to a new base

log a b = log c b log c a (a > 0, a ≠ 1, b > 0, c > 0, c ≠ 1) (8)

That rare case when the ODZ does not change during the conversion. If you have chosen the base c wisely (positive and not equal to 1), the formula for moving to a new base is perfectly safe.

If we choose the number b as a new base c, we obtain an important particular case of formula (8):

Log a b = 1 log b a (a > 0, a ≠ 1, b > 0, b ≠ 1) (9)

Some simple examples with logarithms

Example 1 Calculate: lg2 + lg50.
Decision. lg2 + lg50 = lg100 = 2. We used the formula for the sum of logarithms (5) and the definition of the decimal logarithm.

Example 2 Calculate: lg125/lg5.
Decision. lg125/lg5 = log 5 125 = 3. We used the new base transition formula (8).

Table of formulas related to logarithms

a log a b = b (a > 0, a ≠ 1)

log a a = 1 (a > 0, a ≠ 1)

log a 1 = 0 (a > 0, a ≠ 1)

log a (b c) = log a b + log a c (a > 0, a ≠ 1, b > 0, c > 0)

log a b c = log a b − log a c (a > 0, a ≠ 1, b > 0, c > 0)

log a b p = p log a b (a > 0, a ≠ 1, b > 0)

log a b = log c b log c a (a > 0, a ≠ 1, b > 0, c > 0, c ≠ 1)

log a b = 1 log b a (a > 0, a ≠ 1, b > 0, b ≠ 1)

Portal for the student. Self-training

natural log graph

natural log limit

Formula for changing the base of a logarithm

We solve problems

Examples of solving logarithms

Transition to a new foundation

Basic properties of the logarithm

Basic properties of logarithms

Examples for logarithms

Take the logarithm of expressions

Basic properties of logarithms

Addition and subtraction of logarithms

Removing the exponent from the logarithm

Formulas of logarithms. Logarithms are examples of solutions.

Transition to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Basic properties of the logarithm

Common cases of logarithms

Examples for logarithms

Take the logarithm of expressions

Finding Logarithm Values

Logarithms. First level.

Basic properties of logarithms

Addition and subtraction of logarithms

Removing the exponent from the logarithm

How to solve logarithms

Transition to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Logarithm

Logarithm function and its properties

Examples of solving typical USE problems

Basic logarithmic identity

Two obvious consequences of the definition of the logarithm

The logarithm of the product and the logarithm of the quotient

The degree can be taken out of the sign of the logarithm

Formula for moving to a new base

Some simple examples with logarithms

Table of formulas related to logarithms

RELATED ARTICLES