One of the elements of primitive level algebra is the logarithm. The name comes from the Greek language from the word “number” or “power” and means the power to which the number in the base must be raised to find the final number.
Types of logarithms
- log a b – logarithm of the number b to base a (a > 0, a ≠ 1, b > 0);
- log b – decimal logarithm (logarithm to base 10, a = 10);
- ln b – natural logarithm (logarithm to base e, a = e).
How to solve logarithms?
The logarithm of b to base a is an exponent that requires b to be raised to base a. The result obtained is pronounced like this: “logarithm of b to base a.” The solution to logarithmic problems is that you need to determine the given power in numbers from the specified numbers. There are some basic rules to determine or solve the logarithm, as well as convert the notation itself. Using them, logarithmic equations are solved, derivatives are found, integrals are solved, and many other operations are carried out. Basically, the solution to the logarithm itself is its simplified notation. Below are the basic formulas and properties:
For any a ; a > 0; a ≠ 1 and for any x ; y > 0.
- a log a b = b – basic logarithmic identity
- log a 1 = 0
- loga a = 1
- log a (x y) = log a x + log a y
- log a x/ y = log a x – log a y
- log a 1/x = -log a x
- log a x p = p log a x
- log a k x = 1/k log a x , for k ≠ 0
- log a x = log a c x c
- log a x = log b x/ log b a – formula for moving to a new base
- log a x = 1/log x a
How to solve logarithms - step-by-step instructions for solving
- First, write down the required equation.
Please note: if the base logarithm is 10, then the entry is shortened, resulting in a decimal logarithm. If there is a natural number e, then we write it down, reducing it to a natural logarithm. This means that the result of all logarithms is the power to which the base number is raised to obtain the number b.
Directly, the solution lies in calculating this degree. Before solving an expression with a logarithm, it must be simplified according to the rule, that is, using formulas. You can find the main identities by going back a little in the article.
When adding and subtracting logarithms with two different numbers but with the same bases, replace with one logarithm with the product or division of the numbers b and c, respectively. In this case, you can apply the formula for moving to another base (see above).
If you use expressions to simplify a logarithm, there are some limitations to consider. And that is: the base of the logarithm a is only a positive number, but not equal to one. The number b, like a, must be greater than zero.
There are cases where, by simplifying an expression, you will not be able to calculate the logarithm numerically. It happens that such an expression does not make sense, because many powers are irrational numbers. Under this condition, leave the power of the number as a logarithm.
Logarithm of the number b (b > 0) to base a (a > 0, a ≠ 1)– exponent to which the number a must be raised to obtain b.
The base 10 logarithm of b can be written as log(b), and the logarithm to base e (natural logarithm) is ln(b).
Often used when solving problems with logarithms:
Properties of logarithms
There are four main properties of logarithms.
Let a > 0, a ≠ 1, x > 0 and y > 0.
Property 1. Logarithm of the product
Logarithm of the product 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 equal to the difference of logarithms:
log a (x / y) = log a x – log a y
Property 3. Logarithm of power
Logarithm of degree equal to the product of the power and the logarithm:
If the base of the logarithm is in the degree, then another formula applies:
Property 4. Logarithm of the root
This property can be obtained from the property of the logarithm of a power, since the nth root of the power is equal to the power of 1/n:
Formula for converting from a logarithm in one base to a logarithm in another base
This formula is also often used when solving various tasks on logarithms:
Special case:
Comparing logarithms (inequalities)
Let us have 2 functions f(x) and g(x) under logarithms with the same bases and between them there is an inequality sign:
To compare them, you need to first 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
Problems with logarithms included in the Unified State Examination in mathematics for grade 11 in task 5 and task 7, you can find tasks with solutions on our website in the appropriate sections. Also, tasks with logarithms are found in the math task bank. You can find all examples by searching the site.
What is a logarithm
Logarithms have always been considered a difficult topic in school mathematics courses. There are many different definitions of logarithm, but for some reason most textbooks use the most complex and unsuccessful of them.
We will define the logarithm simply and clearly. To do this, let's create a table:
So, we have powers of two.
Logarithms - properties, formulas, how to solve
If you take the number from the bottom line, you can easily find the power to which you will have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.
And now - actually, the definition of the logarithm:
the base a of the argument x is the power to which the number a must be raised to obtain the number x.
Designation: log a x = b, where a is the base, x is the argument, b is what the logarithm is actually equal to.
For example, 2 3 = 8 ⇒log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). With the same success, 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 line to our table:
| 2 1 | 2 2 | 2 3 | 2 4 | 2 5 | 2 6 |
| 2 | 4 | 8 | 16 | 32 | 64 |
| log 2 2 = 1 | log 2 4 = 2 | log 2 8 = 3 | log 2 16 = 4 | log 2 32 = 5 | log 2 64 = 6 |
Unfortunately, not all logarithms are calculated so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the interval. 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 ad infinitum, and they are never repeated. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100.
It is important to understand that a logarithm is an expression with two variables (the base and the argument). At first, many people confuse where the basis is and where the argument is. To avoid annoying misunderstandings, just look at the picture:
Before us is nothing more than the definition of a logarithm. Remember: logarithm is a power, into which the base must be built in order to obtain an argument. It is the base that is raised to a power - it is highlighted in red in the picture. It turns out that the base is always at the bottom! I tell my students this wonderful rule at the very first lesson - and no confusion arises.
How to count logarithms
We've figured out the definition - all that remains is 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:
- The argument and the base must always be greater than zero. This follows from the definition of a degree by a rational exponent, to which the definition of a logarithm is reduced.
- The base must be different from one, since one to any degree still remains one. 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 range of acceptable values(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒x > 0, a > 0, a ≠ 1.
Note that there are no restrictions on the number b (the value of the logarithm). For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1.
However, now we are considering only numerical expressions, where it is not required to know the VA of the logarithm. All restrictions have already been taken into account by the authors of the problems. But when logarithmic equations and inequalities come into play, DL requirements will become mandatory. After all, the basis and argument may contain very strong constructions that do not necessarily correspond to the above restrictions.
Now let's look at the general scheme for calculating logarithms. It consists of three steps:
- Express the base a and the argument x as a power with the minimum possible base greater than one. Along the way, it’s better to get rid of decimals;
- Solve the equation for variable b: x = a b ;
- The resulting number b will be the answer.
That's all! If the logarithm turns out to be irrational, this will be visible already in the first step. The requirement that the base be greater than one is very important: this reduces the likelihood of error and greatly simplifies the calculations. It’s the same with decimal fractions: if you immediately convert them into ordinary ones, there will be many fewer errors.
Let's see how this scheme works using specific examples:
Task. Calculate the logarithm: log 5 25
- Let's imagine the base and argument as a power of five: 5 = 5 1 ; 25 = 5 2 ;
- We received the answer: 2.
Let's create and solve the equation:
log 5 25 = b ⇒(5 1) b = 5 2 ⇒5 b = 5 2 ⇒ b = 2;
Task. Calculate the logarithm:
Task. Calculate the logarithm: log 4 64
- Let's imagine the base and argument as a power of two: 4 = 2 2 ; 64 = 2 6 ;
- Let's create and solve the equation:
log 4 64 = b ⇒(2 2) b = 2 6 ⇒2 2b = 2 6 ⇒2b = 6 ⇒ b = 3; - We received the answer: 3.
Task. Calculate the logarithm: log 16 1
- Let's imagine the base and argument as a power of two: 16 = 2 4 ; 1 = 2 0 ;
- Let's create and solve the equation:
log 16 1 = b ⇒(2 4) b = 2 0 ⇒2 4b = 2 0 ⇒4b = 0 ⇒ b = 0; - We received the answer: 0.
Task. Calculate the logarithm: log 7 14
- Let's imagine the base and argument as a power of seven: 7 = 7 1 ; 14 cannot be represented as a power of seven, since 7 1< 14 < 7 2 ;
- From the previous paragraph it follows that the logarithm does not count;
- The answer is no change: log 7 14.
A small note on the last example. How can you be sure that a number is not an exact power of another number? It’s very simple - just factor it into prime factors. If the expansion has at least two different factors, the number is not an exact power.
Task. Find out whether the numbers are exact powers: 8; 48; 81; 35; 14.
8 = 2 · 2 · 2 = 2 3 - exact degree, because there is only one multiplier;
48 = 6 · 8 = 3 · 2 · 2 · 2 · 2 = 3 · 2 4 - is not an exact power, since there are two factors: 3 and 2;
81 = 9 · 9 = 3 · 3 · 3 · 3 = 3 4 - exact degree;
35 = 7 · 5 - again not an exact power;
14 = 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 symbol.
of the argument x is the logarithm to base 10, i.e. The power to which the number 10 must be raised to obtain the number x. Designation: lg x.
For example, log 10 = 1; lg 100 = 2; lg 1000 = 3 - etc.
From now on, when a phrase like “Find lg 0.01” appears in a textbook, know that this is not a typo. This is a decimal logarithm. However, if you are unfamiliar with this notation, you can always rewrite it:
log x = log 10 x
Everything that is true for ordinary logarithms is also true for decimal logarithms.
Natural logarithm
There is another logarithm that has its own designation. In some ways, it's even more important than decimal. We are talking about the natural logarithm.
of the argument x is the logarithm to base e, i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x.
Many people will ask: what is the number e? This is an irrational number; its exact value cannot be found and written down. I will give only the first figures:
e = 2.718281828459…
We will not go into detail about what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x
Thus ln e = 1; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, for one: 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 logarithm.
A logarithm is an exponent to which the base must be raised to obtain the number under the logarithm sign.
Thus, in order to represent a certain number c as a logarithm to base a, you need to put a power with the same base as the base of the logarithm under the sign of the logarithm, and write this number c as the exponent:
Absolutely any number can be represented as a logarithm - positive, negative, integer, fractional, rational, irrational:
In order not to confuse a and c under stressful conditions of a test or exam, you can use the following memorization rule:
what is below goes down, what is above goes up.
For example, you need 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, to the base of the degree, and which – up, to the exponent.
The base 3 in the notation of a logarithm is at the bottom, which means that when we represent two as a logarithm to the base 3, we will also write 3 down to the base.
2 is higher than three. And in notation of the degree two we write above the three, that is, as an exponent:
Logarithms. First level.
Logarithms
Logarithm positive number b based on a, Where a > 0, a ≠ 1, is called 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. It is usually called logarithmic identity.
The action of finding the logarithm of a number is called by logarithm.
Properties of logarithms:
Logarithm of the product:
Logarithm of the quotient:
Replacing the logarithm base:
Logarithm of degree:
Logarithm of the root:
Logarithm with power base:
Decimal and natural logarithms.
Decimal logarithm numbers call the logarithm of this number to base 10 and write lg b
Natural logarithm numbers are called the logarithm of that number to the base e, Where e- 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 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: log a x and log a y. Then they can be added and subtracted, and:
- log a x + log a y = log a (x y);
- log a x − log a y = log a (x: y).
So, the sum of 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:
Log 6 4 + log 6 9.
Since logarithms have the same bases, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.
Task. Find the value of the expression: log 2 48 − log 2 3.
The bases are the same, we use the difference formula:
log 2 48 − log 2 3 = log 2 (48: 3) = log 2 16 = 4.
Task. Find the value of the expression: log 3 135 − log 3 5.
Again the bases are the same, so we have:
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.
As you can see, the original expressions are 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.
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.
How to solve logarithms
This is what is most often required.
Task. Find the value of the expression: log 7 49 6 .
Let's get rid of the degree in the argument using the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12
Task. Find the meaning of the expression:
Note that the denominator contains a logarithm, the base and argument of which are exact powers: 16 = 2 4 ; 49 = 7 2. 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.
Now let's look at the main fraction. The numerator and denominator contain 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 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 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 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 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: log 5 16 log 2 25.
Note that the arguments of both logarithms contain exact powers. 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 “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: log 9 100 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: .
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 log 25 64 = log 5 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
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.
- log a a = 1 is. Remember once and for all: the logarithm to any base a of that base itself is equal to one.
- log a 1 = 0 is. The base a can be anything, but if the argument contains one, the logarithm is equal to 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.
As society developed and production became more complex, mathematics also developed. Movement from simple to complex. From ordinary accounting using the method of addition and subtraction, with their repeated repetition, we 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 of multi-digit numbers. The ancient tables were of 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 powers in the form of 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 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 the well-known formula of the 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 is a 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 solving practical problems difficult to implement, we used pre-compiled tables of logarithms, which significantly speeded up all the 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), constructed over several points, allows you to use a regular ruler to find the value of the function at any other point. For a long time, engineers used so-called graph paper for these purposes.
In the 17th century, the first auxiliary analog computing conditions appeared, which by the 19th century acquired a complete form. 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
To solve various equations and inequalities using logarithms, the following formulas are used:
- Transition 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 is greater than one, then the sign of the inequality is preserved; otherwise it changes.
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 for describing objects in the 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 research methods 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 us give only two examples of describing physical laws using the logarithm.
The problem of calculating such a complex quantity as the speed of a rocket can be solved by 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 – 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 the inverse ratio of the stimulus intensity value 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 associated with 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.
Instructions
Write the given logarithmic expression. If the expression uses the logarithm of 10, then its notation is shortened and looks like this: lg b is the decimal logarithm. If the logarithm has the number e as its base, then write the expression: ln b – natural logarithm. It is understood that the result of any is the power to which the base number must be raised to obtain the number b.
When finding the sum of two functions, you simply need to differentiate them one by one and add the results: (u+v)" = u"+v";
When finding the derivative of the product of two functions, it is necessary to multiply the derivative of the first function by the second and add the derivative of the second function multiplied by the first function: (u*v)" = u"*v+v"*u;
In order to find the derivative of the quotient of two functions, it is necessary to subtract from the product of the derivative of the dividend multiplied by the divisor function the product of the derivative of the divisor multiplied by the function of the dividend, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;
If a complex function is given, then it is necessary to multiply the derivative of the internal function and the derivative of the external one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).
Using the results obtained above, you can differentiate almost any function. So let's look at a few examples:
y=x^4, y"=4*x^(4-1)=4*x^3;
y=2*x^3*(e^x-x^2+6), y"=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2 *x));
There are also problems involving calculating the derivative at a point. Let the function y=e^(x^2+6x+5) be given, you need to find the value of the function at the point x=1.
1) Find the derivative of the function: y"=e^(x^2-6x+5)*(2*x +6).
2) Calculate the value of the function at a given point y"(1)=8*e^0=8
Video on the topic
Helpful advice
Learn the table of elementary derivatives. This will significantly save time.
Sources:
- derivative of a constant
So, what is the difference between an irrational equation and a rational one? If the unknown variable is under the square root sign, then the equation is considered irrational.
Instructions
The main method for solving such equations is the method of constructing both sides equations into a square. However. this is natural, the first thing you need to do is get rid of the sign. This method is not technically difficult, but sometimes it can lead to trouble. For example, the equation is v(2x-5)=v(4x-7). By squaring both sides you get 2x-5=4x-7. Solving such an equation is not difficult; x=1. But the number 1 will not be given equations. Why? Substitute one into the equation instead of the value of x. And the right and left sides will contain expressions that do not make sense, that is. This value is not valid for a square root. Therefore, 1 is an extraneous root, and therefore this equation has no roots.
So, an irrational equation is solved using the method of squaring both its sides. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots into the original equation.
Consider another one.
2х+vх-3=0
Of course, this equation can be solved using the same equation as the previous one. Move Compounds equations, which do not have a square root, to the right side and then use the squaring method. solve the resulting rational equation and roots. But also another, more elegant one. Enter a new variable; vх=y. Accordingly, you will receive an equation of the form 2y2+y-3=0. That is, an ordinary quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vх=1; vх=-3/2. The second equation has no roots; from the first we find that x=1. Don't forget to check the roots.
Solving identities is quite simple. To do this, it is necessary to carry out identical transformations until the set goal is achieved. Thus, with the help of simple arithmetic operations, the problem posed will be solved.
You will need
- - paper;
- - pen.
Instructions
The simplest of such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), difference of squares, sum (difference), cube of the sum (difference)). In addition, there are many trigonometric formulas, which are essentially the same identities.
Indeed, the square of the sum of two terms is equal to the square of the first plus twice the product of the first by the second and plus the square of the second, that is, (a+b)^2= (a+b)(a+b)=a^2+ab +ba+b ^2=a^2+2ab+b^2.
Simplify both
General principles of the solution
Repeat from a textbook on mathematical analysis or higher mathematics what a definite integral is. As is known, the solution to a definite integral is a function whose derivative will give an integrand. This function is called antiderivative. Based on this principle, the main integrals are constructed.Determine by the type of the integrand which of the table integrals is suitable in this case. It is not always possible to determine this immediately. Often, the tabular form becomes noticeable only after several transformations to simplify the integrand.
Variable Replacement Method
If the integrand is a trigonometric function whose argument is a polynomial, then try using the change of variables method. In order to do this, replace the polynomial in the argument of the integrand with some new variable. Based on the relationship between the new and old variables, determine the new limits of integration. By differentiating this expression, find the new differential in . Thus, you will get a new form of the previous integral, close or even corresponding to some tabular one.Solving integrals of the second kind
If the integral is an integral of the second kind, a vector form of the integrand, then you will need to use the rules for the transition from these integrals to scalar ones. One such rule is the Ostrogradsky-Gauss relation. This law allows us to move from the rotor flux of a certain vector function to the triple integral over the divergence of a given vector field.Substitution of integration limits
After finding the antiderivative, it is necessary to substitute the limits of integration. First, substitute the value of the upper limit into the expression for the antiderivative. You will get some number. Next, subtract from the resulting number another number obtained from the lower limit into the antiderivative. If one of the limits of integration is infinity, then when substituting it into the antiderivative function, it is necessary to go to the limit and find what the expression tends to.If the integral is two-dimensional or three-dimensional, then you will have to represent the limits of integration geometrically to understand how to evaluate the integral. Indeed, in the case of, say, a three-dimensional integral, the limits of integration can be entire planes that limit the volume being integrated.
Logarithm of a number N based on A called exponent X , to which you need to build A to get the number N
Provided that 
,
,
From the definition of logarithm it follows that 
, i.e.
- this equality is the basic logarithmic identity.
Logarithms based on base 10 are called decimal logarithms. Instead of 
write 
.
Logarithms to the base e
are called natural and are designated 
.
Basic properties of logarithms.
The logarithm of one is equal to zero for any base.
The logarithm of the product is equal to the sum of the logarithms of the factors.
3) The logarithm of the quotient is equal to the difference of the logarithms
Factor 
called the modulus of transition from logarithms to the base a
to logarithms at the base b
.
Using properties 2-5, it is often possible to reduce the logarithm of a complex expression to the result of simple arithmetic operations on logarithms.
For example, 
Such transformations of a logarithm are called logarithms. Transformations inverse to logarithms are called potentiation.
Chapter 2. Elements of higher mathematics.
1. Limits
Limit of the function 
is a finite number A if, as xx
0
for each predetermined 
, there is such a number 
that as soon as 
, That 
.
A function that has a limit differs from it by an infinitesimal amount: 
, where- b.m.v., i.e. 
.
Example. Consider the function 
.
When striving 
, function y
tends to zero: 
1.1. Basic theorems about limits.
The limit of a constant value is equal to this constant value
.
The limit of the sum (difference) of a finite number of functions is equal to the sum (difference) of the limits of these functions.
The limit of the product of a finite number of functions is equal to the product of the limits of these functions.
The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is not zero.
Wonderful Limits
,
, Where 
1.2. Limit Calculation Examples
However, not all limits are calculated so easily. More often, calculating the limit comes down to revealing an uncertainty of the type: 
or .
.
2. Derivative of a function
Let us have a function 
, continuous on the segment 
.
Argument 
got some increase 
. Then the function will receive an increment 
.
Argument value 
corresponds to the function value 
.
Argument value 
corresponds to the function value.
Hence, .
Let us find the limit of this ratio at 
. If this limit exists, then it is called the derivative of the given function.
Definition 3 Derivative of a given function 
by argument 
is called the limit of the ratio of the increment of a function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.
Derivative of a function 
can be designated as follows:
;
;
;
.
Definition 4The operation of finding the derivative of a function is called differentiation.
2.1. Mechanical meaning of derivative.
Let's consider the rectilinear motion of some rigid body or material point.
Let at some point in time 
moving point 
was at a distance 
from the starting position 
.
After some period of time 
she moved a distance 
. Attitude 
=
- average speed of a material point 
. Let us find the limit of this ratio, taking into account that 
.
Consequently, determining the instantaneous speed of movement of a material point is reduced to finding the derivative of the path with respect to time.
2.2. Geometric value of the derivative
Let us have a graphically defined function 
.
Rice. 1. Geometric meaning of derivative
If 
, then point 
, will move along the curve, approaching the point 
.
Hence 
, i.e. the value of the derivative for a given value of the argument 
numerically equal to the tangent of the angle formed by the tangent at a given point with the positive direction of the axis 
.
2.3. Table of basic differentiation formulas.
Power function
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Exponential function
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Logarithmic function
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Trigonometric function
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Inverse trigonometric function
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2.4. Rules of differentiation.
Derivative of 
Derivative of the sum (difference) of functions
Derivative of the product of two functions
Derivative of the quotient of two functions
2.5. Derivative of a complex function.
Let the function be given 
such that it can be represented in the form
And 
, where the variable 
is an intermediate argument, then 
The derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument with respect to x.
Example 1. 
Example 2. 
3. Differential function.
Let there be 
, differentiable on some interval 
let it go at
this function has a derivative
,
then we can write
(1),
Where 
- an infinitesimal quantity,
since when 
Multiplying all terms of equality (1) by 
we have:
Where 
- b.m.v. higher order.
Magnitude 
called the differential of the function 
and is designated 
.
3.1. Geometric value of the differential.
Let the function be given 
.
Fig.2. Geometric meaning of differential.
.
Obviously, the differential of the function 
is equal to the increment of the ordinate of the tangent at a given point.
3.2. Derivatives and differentials of various orders.
If there 
, Then 
is called the first derivative.
The derivative of the first derivative is called the second-order derivative and is written 
.
Derivative of the nth order of the function 
is called the (n-1)th order derivative and is written:
.
The differential of the differential of a function is called the second differential or second order differential.
.
.
3.3 Solving biological problems using differentiation.
Task 1. Studies have shown that the growth of a colony of microorganisms obeys the law 
, Where N
– number of microorganisms (in thousands), t
– time (days).
b) Will the population of the colony increase or decrease during this period?
Answer. The size of the colony will increase.
Task 2. The water in the lake is periodically tested to monitor the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio
.
When will the lake have a minimum concentration of bacteria and will it be possible to swim in it?
Solution: A function reaches max or min when its derivative is zero.
,
Let's determine the max or min will be in 6 days. To do this, let's take the second derivative.
Answer: After 6 days there will be a minimum concentration of bacteria.