How do logarithms add up. Properties of logarithms and examples of their solutions

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The logarithm of a number N by reason a is called exponent X , to which you need to raise a to get the number N

Provided that
,
,

It follows from the definition of the logarithm that
, i.e.
- this equality is the basic logarithmic identity.

Logarithms to base 10 are called decimal logarithms. Instead of
write
.

base logarithms e are called natural and denoted
.

Basic properties of logarithms.

The logarithm of unity for any base is zero

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
is called the modulus of transition from logarithms at 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 the logarithm are called logarithms. Transformations reciprocal of logarithms are called potentiation.

Chapter 2. Elements of higher mathematics.

1. Limits

function limit
is a finite number A if, when striving xx 0 for each predetermined
, there is a number
that as soon as
, then
.

A function that has a limit differs from it by an infinitesimal amount:
, where - b.m.w., i.e.
.

Example. Consider the function
.

When striving
, function y goes 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 a 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 equal to zero.

Remarkable Limits

,
, where

1.2. Limit Calculation Examples

However, not all limits are calculated so simply. More often, the calculation of the limit is reduced to the disclosure of type uncertainty: or .

.

2. Derivative of a function

Let we have a function
, continuous on the segment
.

Argument got some boost
. Then the function will be incremented
.

Argument value corresponds to the value of the function
.

Argument value
corresponds to the value of the function .

Consequently, .

Let us find the limit of this relation at
. If this limit exists, then it is called the derivative of the given function.

Definition of the 3derivative of a given function
by argument called the limit of the ratio of the increment of the function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.

Function derivative
can be denoted as follows:

; ; ; .

Definition 4The operation of finding the derivative of a function is called differentiation.

2.1. The mechanical meaning of the derivative.

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 material point
. Let us find the limit of this ratio, taking into account that
.

Hence the definition instantaneous speed the motion of a material point is reduced to finding the derivative of the path with respect to time.

2.2. geometric value derivative

Suppose we have a graphically defined some function
.

Rice. 1. The geometric meaning of the derivative

If a
, then the point
, will move along the curve, approaching the point
.

Consequently
, i.e. the value of the derivative given the value of the argument numerically equals 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

Exponential function

logarithmic function

trigonometric function

Reverse trigonometric function

2.4. Differentiation rules.

Derivative of

Derivative of the sum (difference) of functions

Derivative of the product of two functions

The derivative of the quotient of two functions

2.5. Derivative of a complex function.

Let the function
such that it can be represented as

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 by the derivative of the intermediate argument with respect to x.

Example1.

Example2.

3. Function differential.

Let there be
, differentiable on some interval
let it go at this function has a derivative

,

then you can write

(1),

where - an infinitesimal quantity,

because at

Multiplying all terms of equality (1) by
we have:

Where
- b.m.v. higher order.

Value
is called the differential of the function
and denoted

.

3.1. The geometric value of the differential.

Let the function
.

Fig.2. The geometric meaning of the differential.

.

Obviously, the differential of the function
is equal to the increment of the ordinate of the tangent at the given point.

3.2. Derivatives and differentials of various orders.

If there is
, 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 derivative of the (n-1) order and is written:

.

The differential of the differential of a function is called the second differential or the second order differential.

.

.

3.3 Solving biological problems using differentiation.

Task1. 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 colony will grow in size.

Task 2. The water in the lake is periodically tested to control the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio

.

When will the minimum concentration of bacteria come in the lake and it will be possible to swim in it?

Solution A function reaches max or min when its derivative is zero.

,

Let's determine max or min will be in 6 days. To do this, we take the second derivative.

Answer: After 6 days there will be a minimum concentration of bacteria.

1. Check if there are negative numbers or one under the logarithm sign. This method applicable to expressions of the form log b ⁡ (x) log b ⁡ (a) (\displaystyle (\frac (\log _(b)(x))(\log _(b)(a)))). However, it is not suitable for some special cases:

   • Logarithm negative number undefined for any reason (e.g., log ⁡ (− 3) (\displaystyle \log(-3)) or log 4 ⁡ (− 5) (\displaystyle \log _(4)(-5))). In this case, write "no solution".
   • The logarithm of zero to any base is also undefined. If you got caught ln ⁡ (0) (\displaystyle \ln(0)), write "no solution".
   • The logarithm of unity in any base ( log ⁡ (1) (\displaystyle \log(1))) is always zero because x 0 = 1 (\displaystyle x^(0)=1) for all values x. Write instead of such a logarithm 1 and do not use the method below.
   • If logarithms have different grounds, for example l o g 3 (x) l o g 4 (a) (\displaystyle (\frac (log_(3)(x))(log_(4)(a)))), and are not reduced to integers, the value of the expression cannot be found manually.

2. Convert the expression to one logarithm. If the expression is not one of the above special occasions, it can be represented as a single logarithm. Use the following formula for this: log b ⁡ (x) log b ⁡ (a) = log a ⁡ (x) (\displaystyle (\frac (\log _(b)(x))(\log _(b)(a)))=\ log_(a)(x)).

   • Example 1: consider the expression log ⁡ 16 log ⁡ 2 (\displaystyle (\frac (\log (16))(\log (2)))).
     First, let's represent the expression as a single logarithm using the above formula: log ⁡ 16 log ⁡ 2 = log 2 ⁡ (16) (\displaystyle (\frac (\log (16))(\log (2)))=\log _(2)(16)).
   • This "change of base" formula for the logarithm is derived from the basic properties of logarithms.

3. If possible, calculate the value of the expression manually. To find log a ⁡ (x) (\displaystyle \log _(a)(x)), imagine the expression " a? = x (\displaystyle a^(?)=x)", that is, ask the following question:" To what power it is necessary to raise a, To obtain x?". This question may require a calculator, but if you're lucky, you can find it manually.

   • Example 1 (continued): Rewrite as 2? = 16 (\displaystyle 2^(?)=16). It is necessary to find what number should stand instead of the sign "?". This can be done by trial and error:
     2 2 = 2 ∗ 2 = 4 (\displaystyle 2^(2)=2*2=4)
     2 3 = 4 ∗ 2 = 8 (\displaystyle 2^(3)=4*2=8)
     2 4 = 8 ∗ 2 = 16 (\displaystyle 2^(4)=8*2=16)
     So, the desired number is 4: log 2 ⁡ (16) (\displaystyle \log _(2)(16)) = 4 .

4. Leave the answer in logarithmic form if you can't simplify it. Many logarithms are very difficult to calculate by hand. In this case, you will need a calculator to get an accurate answer. However, if you are solving a problem in class, then the teacher will most likely be satisfied with the answer in logarithmic form. The method below is used to solve a more complex example:

   • example 2: what is equal log 3 ⁡ (58) log 3 ⁡ (7) (\displaystyle (\frac (\log _(3)(58))(\log _(3)(7))))?
   • Let's convert this expression to one logarithm: log 3 ⁡ (58) log 3 ⁡ (7) = log 7 ⁡ (58) (\displaystyle (\frac (\log _(3)(58))(\log _(3)(7)))=\ log_(7)(58)). Note that the base 3 common to both logarithms disappears; this is true for any base.
   • Let's rewrite the expression in the form 7? = 58 (\displaystyle 7^(?)=58) and try to find the value?:
     7 2 = 7 ∗ 7 = 49 (\displaystyle 7^(2)=7*7=49)
     7 3 = 49 ∗ 7 = 343 (\displaystyle 7^(3)=49*7=343)
     Since 58 is between these two numbers, it is not expressed as a whole number.
   • We leave the answer in logarithmic form: log 7 ⁡ (58) (\displaystyle \log _(7)(58)).

Instruction

Write down 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 the base, then the expression is written: ln b is the natural logarithm. It is understood that the result of any is the power to which the base number must be raised to get the number b.

When finding the sum of two functions, you just 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, from the product of the derivative of the dividend multiplied by the divisor function, to subtract the product of the derivative of the divisor multiplied by the divisor function, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;

If given complex function, then it is necessary to multiply the derivative of internal function and the derivative of the outer one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).

Using the 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 tasks for 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 in given point y"(1)=8*e^0=8

Related videos

Useful advice

Learn the table of elementary derivatives. This will save a lot of time.

Sources:

• constant derivative

So what is the difference between an irrational equation and a rational one? If the unknown variable is under the sign square root, then the equation is considered irrational.

Instruction

The main method for solving such equations is the method of raising both sides equations into a square. However. this is natural, the first step is to get rid of the sign. Technically, this method is not difficult, but sometimes it can lead to trouble. For example, the equation v(2x-5)=v(4x-7). By squaring both sides, you get 2x-5=4x-7. Such an equation is not difficult to solve; x=1. But the number 1 will not be given equations. Why? Substitute the unit in the equation instead of the x value. And the right and left sides will contain expressions that do not make sense, that is. Such a value is not valid for a square root. Therefore, 1 is an extraneous root, and therefore this equation has no roots.

So, the irrational equation is solved using the method of squaring both of its parts. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots in the original equation.

Consider another one.
2x+vx-3=0
Of course, this equation can be solved using the same equation as the previous one. Transfer 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 another, more elegant one. Enter a new variable; vx=y. Accordingly, you will get an equation like 2y2+y-3=0. That is the usual quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vx=1; vx \u003d -3/2. The second equation has no roots, from the first we find that x=1. Do not forget about the need to check the roots.

Solving identities is quite easy. This requires making identical transformations until the goal is achieved. Thus, with the help of the simplest arithmetic operations, the task will be solved.

You will need

• - paper;
• - a pen.

Instruction

The simplest such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), the difference of squares, the sum (difference), the 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 and the second 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 solution

Variable substitution method

Solution of integrals of the second kind

Substitution of limits of integration

With the development of society, the complexity of production, mathematics also developed. Movement from simple to complex. From the usual accounting method of addition and subtraction, with their repeated repetition, they came to the concept of multiplication and division. The reduction of the multiply repeated operation became the concept of exponentiation. The first tables of the dependence of numbers on the base and the number of exponentiation were compiled back in the 8th century by the Indian mathematician Varasena. From them, you can count the time of occurrence of logarithms.

Historical outline

The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation associated with multiplication and division of multi-digit numbers. The ancient tables did a great service. They allowed to replace complex operations to simpler ones - addition and subtraction. A big step forward was the work of the mathematician Michael Stiefel, published in 1544, in which he realized the idea of ​​many mathematicians. This made it possible to use tables not only for degrees in the form prime numbers, but also for arbitrary rational ones.

In 1614, the Scotsman John Napier, developing these ideas, first introduced the new term "logarithm of a number." New complex tables were compiled for calculating the logarithms of sines and cosines, as well as tangents. This greatly reduced the work of astronomers.

New tables began to appear, which were successfully used by scientists for three centuries. A lot of time passed before the new operation in algebra acquired its finished form. The logarithm was defined and its properties were studied.

Only in the 20th century, with the advent of the calculator and the computer, mankind abandoned the ancient tables that had been successfully operating throughout the 13th centuries.

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

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

Thus, the formulated definition puts only one restriction, the numbers a and b must be real.

Varieties of logarithms

The classical definition is called the real logarithm and is actually a solution to the equation a x = b. The option a = 1 is borderline and is of no interest. Note: 1 to any power is 1.

Real value of the logarithm defined only if the base and the argument is greater than 0, and the base must not be equal to 1.

Special place in the field of mathematics play logarithms, which will be named depending on the value of their base:

Rules and restrictions

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

As a variant of this statement, it will be: log c (b / p) \u003d log c (b) - log c (p), the quotient function is equal to the difference of the functions.

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

Other properties include:

Comment. Do not make a common mistake - the logarithm of the sum is not equal to the sum of the logarithms.

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

ln (1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... + ((-1)^(n + 1))*(( x^n)/n), where n is natural number greater than 1, which determines the accuracy of the calculation.

Logarithms with other bases were calculated using the theorem on the transition from one base to another and the property of the logarithm of the product.

Since this method is very laborious and when solving practical problems difficult to implement, they used pre-compiled tables of logarithms, which greatly accelerated the entire work.

In some cases, specially compiled graphs of logarithms were used, which gave less accuracy, but significantly speeded up the search for the desired value. The curve of the function y = log a(x), built on several points, allows using the usual ruler to find the values ​​of the function at any other point. Engineers long time for these purposes, the so-called graph paper was used.

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

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

Equations and inequalities

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

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

To solve inequalities, it is useful to know:

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

Task examples

Consider several options for using logarithms and their properties. Examples with solving equations:

Consider the option of placing the logarithm in the degree:

• Task 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the notation is similar to the following (5^2)^log5(3) or 5^(2 * log 5(3)). Let's write it differently: 5^log 5(3*2), or the square of a number as a function argument can be written as the square of the function itself (5^log 5(3))^2. Using the properties of logarithms, this expression is 3^2. Answer: as a result of the calculation we get 9.

Practical use

Being a purely mathematical tool, it seems far from real life that the logarithm suddenly acquired great importance to describe objects real world. It is difficult to find a science where it is not used. This fully applies not only to the natural, but also to the humanities fields of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical methods research and at the same time served as a stimulus for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. We give only two examples of describing physical laws using the logarithm.

It is possible to solve the problem of calculating such a complex quantity as the speed of a rocket using the Tsiolkovsky formula, which laid the foundation for the theory of space exploration:

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

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

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

S = k * ln (Ω), where

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

Chemistry

Less obvious would be the use of formulas in chemistry containing the ratio of logarithms. Here are just two examples:

• The Nernst equation, the condition of the redox potential of the medium in relation to the activity of substances and the equilibrium constant.
• The calculation of such constants as the autoprolysis index and the acidity of the solution is also not complete without our function.

Psychology and biology

And it’s completely incomprehensible what the psychology has to do with it. It turns out that the strength of sensation is well described by this function as the inverse ratio of the intensity value of the stimulus to the lower intensity value.

After the above examples, it is no longer surprising that the theme of logarithms is also widely used in biology. Whole volumes can be written about biological forms corresponding to logarithmic spirals.

Other areas

It seems that the existence of the world is impossible without connection with this function, and it governs all laws. Especially when the laws of nature are connected with geometric progression. It is worth referring to the MatProfi website, and there are many such examples in the following areas of activity:

The list could be endless. Having mastered the basic laws of this function, you can plunge into the world of infinite wisdom.