Graph of the function logarithm 2 to base x. What is a logarithm? Why are logarithms needed? Domain, set of values, ascending, descending
LOGARITHM
a number that simplifies many complex arithmetic operations. Using their logarithms instead of numbers in calculations makes it possible to replace multiplication with a simpler operation of addition, division with subtraction, raising to a power with multiplication, and extracting roots with division. General description. The logarithm of a given number is the exponent to which another number, called the base of the logarithm, must be raised to get the given number. For example, the base 10 logarithm of 100 is 2. In other words, 10 must be squared to get 100 (102 = 100). If n is a given number, b is a base, and l is a logarithm, then bl = n. The number n is also called the antilogarithm to the base b of the number l. For example, the antilogarithm of 2 to base 10 is 100. This can be written as logb n = l and antilogb l = n. The main properties of logarithms:
Any positive number other than one can serve as the base of logarithms, but, unfortunately, it turns out that if b and n are rational numbers, then in rare cases there is a rational number l such that bl = n. However, it is possible to define an irrational number l, for example, such that 10l = 2; this irrational number l can be approximated by rational numbers with any required accuracy. It turns out that in the example above, l is approximately equal to 0.3010, and this approximate value of the logarithm to the base 10 of the number 2 can be found in four-digit tables of decimal logarithms. Base 10 logarithms (or decimal logarithms) are used so often in calculations that they are called ordinary logarithms and are written as log2 = 0.3010 or log2 = 0.3010, omitting the explicit indication of the base of the logarithm. Logarithms to the base e, a transcendental number approximately equal to 2.71828, are called natural logarithms. They are found mainly in works on mathematical analysis and its applications to various sciences. Natural logarithms are also written without explicitly indicating the base, but using the special notation ln: for example, ln2 = 0.6931, because e0.6931 = 2.
see also NUMBER e . Using tables of ordinary logarithms. The ordinary logarithm of a number is the exponent to which you need to raise 10 to get the given number. Since 100 = 1, 101 = 10, and 102 = 100, we immediately get that log1 = 0, log10 = 1, log100 = 2, and so on. for increasing integer powers of 10. Similarly, 10-1 = 0.1, 10-2 = 0.01 and hence log0.1 = -1, log0.01 = -2, and so on. for all negative integer powers of 10. The usual logarithms of the remaining numbers are enclosed between the logarithms of the nearest integer powers of 10; log2 must be enclosed between 0 and 1, log20 between 1 and 2, and log0.2 between -1 and 0. Thus, the logarithm has two parts, an integer and a decimal enclosed between 0 and 1. The integer part is called the characteristic of the logarithm and is determined by the number itself, the fractional part is called the mantissa and can be found from the tables. Also, log20 = log(2´10) = log2 + log10 = (log2) + 1. The logarithm of 2 is 0.3010, so log20 = 0.3010 + 1 = 1.3010. Similarly, log0.2 = log(2e10) = log2 - log10 = (log2) - 1 = 0.3010 - 1. By subtracting, we get log0.2 = - 0.6990. However, it is more convenient to represent log0.2 as 0.3010 - 1 or as 9.3010 - 10; a general rule can also be formulated: all numbers obtained from a given number by multiplying by a power of 10 have the same mantissa equal to the mantissa of a given number. In most tables, the mantissas of numbers ranging from 1 to 10 are given, since the mantissas of all other numbers can be obtained from those given in the table. Most tables give logarithms to four or five decimal places, although there are seven-digit tables and tables with even more decimal places. Learning how to use such tables is easiest with examples. To find log3.59, first of all, note that the number 3.59 is between 100 and 101, so its characteristic is 0. We find the number 35 in the table (on the left) and move along the row to the column that has the number 9 on top; the intersection of this column and row 35 is 5551, so log3.59 = 0.5551. To find the mantissa of a number with four significant digits, you need to resort to interpolation. In some tables, interpolation is facilitated by the proportional parts given in the last nine columns on the right side of each table page. Find now log736.4; the number 736.4 lies between 102 and 103, so the characteristic of its logarithm is 2. In the table we find the row to the left of which is 73 and column 6. At the intersection of this row and this column is the number 8669. Among the linear parts we find column 4. At the intersection of row 73 and column 4 is the number 2. Adding 2 to 8669, we get the mantissa - it is equal to 8671. Thus, log736.4 = 2, 8671.
natural logarithms. Tables and properties of natural logarithms are similar to tables and properties of ordinary logarithms. The main difference between the two is that the integer part of the natural logarithm is not significant in determining the position of the decimal point, and therefore the difference between the mantissa and the characteristic does not play a special role. Natural logarithms of numbers 5.432; 54.32 and 543.2 are, respectively, 1.6923; 3.9949 and 6.2975. The relationship between these logarithms becomes apparent if we consider the differences between them: log543.2 - log54.32 = 6.2975 - 3.9949 = 2.3026; the last number is nothing but the natural logarithm of the number 10 (written like this: ln10); log543.2 - log5.432 = 4.6052; the last number is 2ln10. But 543.2 = 10*54.32 = 102*5.432. Thus, by the natural logarithm of a given number a, one can find the natural logarithms of numbers equal to the products of the number a and any powers of n of the number 10, if ln10 multiplied by n is added to lna, i.e. ln(a*10n) = lna + nln10 = lna + 2.3026n. For example, ln0.005432 = ln(5.432*10-3) = ln5.432 - 3ln10 = 1.6923 - (3*2.3026) = - 5.2155. Therefore, tables of natural logarithms, like tables of ordinary logarithms, usually contain only the logarithms of numbers from 1 to 10. In the system of natural logarithms, one can speak of antilogarithms, but more often one speaks of an exponential function or an exponential. If x = lny, then y = ex, and y is called the exponent of x (for typographical convenience, y = exp x is often written). The exponent plays the role of the antilogarithm of the number x. Using tables of decimal and natural logarithms, you can create tables of logarithms in any base other than 10 and e. If logb a = x, then bx = a, and hence logc bx = logc a or xlogc b = logc a, or x = logc a/logc b = logb a. Therefore, using this inversion formula from a table of logarithms to base c, one can construct tables of logarithms to any other base b. The factor 1/logc b is called the modulus of the transition from base c to base b. Nothing prevents, for example, using the inversion formula, or the transition from one system of logarithms to another, to find natural logarithms from the table of ordinary logarithms or to make the reverse transition. For example, log105.432 = loge 5.432/loge 10 = 1.6923/2.3026 = 1.6923 x 0.4343 = 0.7350. The number 0.4343, by which the natural logarithm of a given number must be multiplied to obtain the ordinary logarithm, is the modulus of the transition to the system of ordinary logarithms.
Special tables. Logarithms were originally invented to use their properties logab = loga + logb and loga/b = loga - logb to convert products into sums and quotients into differences. In other words, if loga and logb are known, then with the help of addition and subtraction we can easily find the logarithm of the product and the quotient. In astronomy, however, it is often necessary to find log(a + b) or log(a - b) given values of loga and logb. Of course, it would be possible to first find a and b from the tables of logarithms, then perform the indicated addition or subtraction and, again referring to the tables, find the required logarithms, but such a procedure would require three visits to the tables. Z. Leonelli in 1802 published tables of the so-called. Gaussian logarithms - the logarithms of addition of sums and differences - which made it possible to confine ourselves to one recourse to tables. In 1624, I. Kepler proposed tables of proportional logarithms, i.e. logarithms of numbers a/x, where a is some positive constant. These tables are used primarily by astronomers and navigators. Proportional logarithms for a = 1 are called logarithms and are used in calculations when you have to deal with products and quotients. The logarithm of the number n is equal to the logarithm of the reciprocal of the number; those. cologn = log1/n = - logn. If log2 = 0.3010, then colog2 = - 0.3010 = 0.6990 - 1. The advantage of using logarithms is that when calculating the value of the logarithm of expressions like pq/r, the triple sum of the positive decimals of logp + logq + cologr is easier to find than the mixed sum and difference logp + logq - logr.
Story. The principle underlying any system of logarithms has been known for a very long time and can be traced back to ancient Babylonian mathematics (circa 2000 BC). In those days, interpolation between tabular values of positive integer powers was used to calculate compound interest. Much later, Archimedes (287-212 BC) used the powers of 108 to find an upper limit on the number of grains of sand needed to completely fill the universe known at that time. Archimedes drew attention to the property of the exponents that underlies the effectiveness of logarithms: the product of the powers corresponds to the sum of the exponents. At the end of the Middle Ages and the beginning of the New Age, mathematicians increasingly began to refer to the relationship between geometric and arithmetic progressions. M. Stiefel in his essay Arithmetic of Integers (1544) gave a table of positive and negative powers of the number 2:
Stiefel noticed that the sum of the two numbers in the first row (the row of exponents) is equal to the exponent of two, which corresponds to the product of the two corresponding numbers in the bottom row (the row of exponents). In connection with this table, Stiefel formulated four rules that are equivalent to the four modern rules for operations on exponents or four rules for operations on logarithms: the sum in the top row corresponds to the product in the bottom row; the subtraction in the top row corresponds to the division in the bottom row; multiplication in the top row corresponds to exponentiation in the bottom row; the division in the top row corresponds to the root extraction in the bottom row. Apparently, rules similar to those of Stiefel led J. Napier to formally introduce the first system of logarithms in the Description of the amazing table of logarithms, published in 1614. But Napier's thoughts have been occupied with the problem of converting products into sums since more than Ten years before the publication of his work, Napier received news from Denmark that at Tycho Brahe's observatory his assistants had a method for converting products into sums. The method mentioned in Napier's communication was based on the use of trigonometric formulas of the type
Therefore Napier's tables consisted mainly of the logarithms of trigonometric functions. Although the concept of base was not explicitly included in the definition proposed by Napier, the number equivalent to the base of the system of logarithms in his system was played by the number (1 - 10-7)ґ107, approximately equal to 1/e. Independently of Napier and almost simultaneously with him, a system of logarithms, quite similar in type, was invented and published by J. Burgi in Prague, who published the Tables of Arithmetic and Geometric Progressions in 1620. These were tables of antilogarithms in base (1 + 10-4)*10 4, a fairly good approximation of the number e. In Napier's system, the logarithm of the number 107 was taken as zero, and as the numbers decreased, the logarithms increased. When G. Briggs (1561-1631) visited Napier, both agreed that it would be more convenient to use the number 10 as the base and consider the logarithm of one equal to zero. Then, as the numbers increase, their logarithms would increase. Thus we got the modern system of decimal logarithms, a table of which Briggs published in his work Logarithmic Arithmetic (1620). Logarithms to the base e, although not exactly those introduced by Napier, are often called non-Pier. The terms "characteristic" and "mantissa" were proposed by Briggs. The first logarithms, for historical reasons, used approximations to the numbers 1/e and e. Somewhat later, the idea of natural logarithms was associated with the study of areas under the hyperbola xy = 1 (Fig. 1). In the 17th century it has been shown that the area bounded by this curve, the x-axis, and the ordinates x = 1 and x = a (in Fig. 1 this area is covered with thicker and rarer dots) increases exponentially when a increases exponentially. It is this dependence that arises in the rules for actions on exponents and logarithms. This gave grounds to call the Napier logarithms "hyperbolic logarithms".
Logarithmic function. There was a time when logarithms were considered solely as a means of calculation, but in the 18th century, mainly due to the work of Euler, the concept of a logarithmic function was formed. The graph of such a function y = lnx, whose ordinates increase in arithmetic progression, while the abscissas increase in geometric progression, is shown in Fig. 2a. The graph of the inverse, or exponential (exponential) function y = ex, whose ordinates increase exponentially, and the abscissas - arithmetic, is presented, respectively, in Fig. 2b. (The curves y = logx and y = 10x are similar in shape to the curves y = lnx and y = ex.) Alternative definitions of the logarithmic function have also been proposed, for example,
Thanks to Euler's work, the relationships between logarithms and trigonometric functions in the complex plane became known. From the identity eix = cos x + i sin x (where the angle x is measured in radians), Euler concluded that every non-zero real number has infinitely many natural logarithms; they are all complex for negative numbers, and all but one for positive numbers. Since eix = 1 not only for x = 0, but also for x = ± 2kp, where k is any positive integer, any of the numbers 0 ± 2kpi can be taken as the natural logarithm of the number 1; and, similarly, the natural logarithms of -1 are complex numbers of the form (2k + 1)pi, where k is an integer. Similar statements are also true for general logarithms or other systems of logarithms. In addition, the definition of logarithms can be generalized using the Euler identities to include the complex logarithms of complex numbers. An alternative definition of the logarithmic function is provided by functional analysis. If f(x) is a continuous function of a real number x having the following three properties: f(1) = 0, f(b) = 1, f(uv) = f(u) + f(v), then f(x ) is defined as the logarithm of the number x to the base b. This definition has a number of advantages over the definition given at the beginning of this article.
Applications. Logarithms were originally used solely to simplify calculations, and this application is still one of their most important. The calculation of products, quotients, powers and roots is facilitated not only by the wide availability of published tables of logarithms, but also by the use of the so-called. slide rule - a computing tool, the principle of which is based on the properties of logarithms. The ruler is equipped with logarithmic scales, i.e. the distance from the number 1 to any number x is chosen to be log x; by shifting one scale relative to another, it is possible to plot the sums or differences of logarithms, which makes it possible to read products or partials of the corresponding numbers directly from the scale. To take advantage of the presentation of numbers in a logarithmic form allows the so-called. logarithmic paper for plotting (paper with logarithmic scales printed on it along both coordinate axes). If a function satisfies a power law of the form y = kxn, then its logarithmic graph looks like a straight line, because log y = log k + n log x is an equation linear in log y and log x. On the contrary, if the logarithmic graph of some functional dependence has the form of a straight line, then this dependence is a power law. Semi-logarithmic paper (where the y-axis is on a logarithmic scale and the abscissa is on a uniform scale) is useful when exponential functions need to be identified. Equations of the form y = kbrx arise whenever a quantity, such as population, radioactive material, or bank balance, decreases or increases at a rate proportional to the current population, radioactive material, or money. If such a dependence is applied to semi-logarithmic paper, then the graph will look like a straight line. The logarithmic function arises in connection with a variety of natural forms. Flowers in sunflower inflorescences line up in logarithmic spirals, the shells of the Nautilus mollusk, the horns of the mountain sheep and the beaks of parrots are twisted. All of these natural forms are examples of the curve known as the logarithmic spiral because its equation in polar coordinates is r = aebq, or lnr = lna + bq. Such a curve is described by a moving point, the distance from the pole of which grows exponentially, and the angle described by its radius vector grows arithmetic. The ubiquity of such a curve, and consequently of the logarithmic function, is well illustrated by the fact that it occurs in regions as far away and quite different as the contour of the eccentric cam and the trajectory of certain insects flying towards the light.
Collier Encyclopedia. - Open society. 2000 .
See what "LOGARIFM" is in other dictionaries:
- (Greek, from logos relation, and arithmos number). The number of an arithmetic progression corresponding to the number of a geometric progression. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. LOGARIFM Greek, from logos, relation, ... ... Dictionary of foreign words of the Russian language
The given number N at the base a is the exponent of the power of y to which you need to raise the number a to get N; thus, N = ay. The logarithm is usually denoted by logaN. Logarithm with base e? 2.718... is called natural and denoted by lnN.… … Big Encyclopedic Dictionary
- (from the Greek logos ratio and arithmos number) numbers N in base a (O ... Modern Encyclopedia
Acceptable range (ODZ) of the logarithm
Now let's talk about restrictions (ODZ - the area of admissible values of variables).
We remember that, for example, the square root cannot be taken from negative numbers; or if we have a fraction, then the denominator cannot be equal to zero. There are similar restrictions for logarithms:
That is, both the argument and the base must be greater than zero, and the base cannot be equal.
Why is that?
Let's start simple: let's say that. Then, for example, the number does not exist, since no matter what degree we raise, it always turns out. Moreover, it does not exist for any. But at the same time it can be equal to anything (for the same reason - it is equal to any degree). Therefore, the object is of no interest, and it was simply thrown out of mathematics.
We have a similar problem in the case: in any positive degree - this, but it cannot be raised to a negative power at all, since division by zero will result (I remind you that).
When we are faced with the problem of raising to a fractional power (which is represented as a root:. For example, (that is), but does not exist.
Therefore, negative reasons are easier to throw away than to mess with them.
Well, since the base a is only positive for us, then no matter what degree we raise it, we will always get a strictly positive number. So the argument must be positive. For example, it does not exist, since it will not be a negative number to any extent (and even zero, therefore it does not exist either).
In problems with logarithms, the first step is to write down the ODZ. I'll give an example:
Let's solve the equation.
Recall the definition: the logarithm is the power to which the base must be raised in order to obtain an argument. And by the condition, this degree is equal to: .
We get the usual quadratic equation: . We solve it using the Vieta theorem: the sum of the roots is equal, and the product. Easy to pick up, these are numbers and.
But if you immediately take and write down both of these numbers in the answer, you can get 0 points for the task. Why? Let's think about what happens if we substitute these roots into the initial equation?
This is clearly false, since the base cannot be negative, that is, the root is "third-party".
To avoid such unpleasant tricks, you need to write down the ODZ even before starting to solve the equation:
Then, having received the roots and, we immediately discard the root, and write the correct answer.
Example 1(try to solve it yourself) :
Find the root of the equation. If there are several roots, indicate the smaller one in your answer.
Solution:
First of all, let's write the ODZ:
Now we remember what a logarithm is: to what power do you need to raise the base to get an argument? In the second. That is:
It would seem that the smaller root is equal. But this is not so: according to the ODZ, the root is third-party, that is, it is not the root of this equation at all. Thus, the equation has only one root: .
Answer: .
Basic logarithmic identity
Recall the definition of a logarithm in general terms:
Substitute in the second equality instead of the logarithm:
This equality is called basic logarithmic identity. Although in essence this equality is just written differently definition of the logarithm:
This is the power to which you need to raise in order to get.
For example:
Solve the following examples:
Example 2
Find the value of the expression.
Solution:
Recall the rule from the section:, that is, when raising a degree to a power, the indicators are multiplied. Let's apply it:
Example 3
Prove that.
Solution:
Properties of logarithms
Unfortunately, the tasks are not always so simple - often you first need to simplify the expression, bring it to the usual form, and only then it will be possible to calculate the value. It's easiest to do this knowing properties of logarithms. So let's learn the basic properties of logarithms. I will prove each of them, because any rule is easier to remember if you know where it comes from.
All these properties must be remembered; without them, most problems with logarithms cannot be solved.
And now about all the properties of logarithms in more detail.
Property 1:
Proof:
Let, then.
We have: , h.t.d.
Property 2: Sum of logarithms
The sum of logarithms with the same base is equal to the logarithm of the product: .
Proof:
Let, then. Let, then.
Example: Find the value of the expression: .
Solution: .
The formula you just learned helps to simplify the sum of the logarithms, not the difference, so that these logarithms cannot be combined right away. But you can do the opposite - "break" the first logarithm into two: And here is the promised simplification:
.
Why is this needed? Well, for example: what does it matter?
Now it's obvious that.
Now make it easy for yourself:
Tasks:
Answers:
Property 3: Difference of logarithms:
Proof:
Everything is exactly the same as in paragraph 2:
Let, then.
Let, then. We have:
The example from the last point is now even simpler:
More complicated example: . Guess yourself how to decide?
Here it should be noted that we do not have a single formula about logarithms squared. This is something akin to an expression - this cannot be simplified right away.
Therefore, let's digress from the formulas about logarithms, and think about what formulas we generally use in mathematics most often? Ever since 7th grade!
It - . You have to get used to the fact that they are everywhere! And in exponential, and in trigonometric, and in irrational problems, they are found. Therefore, they must be remembered.
If you look closely at the first two terms, it becomes clear that this is difference of squares:
Answer to check:
Simplify yourself.
Examples
Answers.
Property 4: Derivation of the exponent from the argument of the logarithm:
Proof: And here we also use the definition of the logarithm: let, then. We have: , h.t.d.
You can understand this rule like this:
That is, the degree of the argument is taken forward of the logarithm, as a coefficient.
Example: Find the value of the expression.
Solution: .
Decide for yourself:
Examples:
Answers:
Property 5: Derivation of the exponent from the base of the logarithm:
Proof: Let, then.
We have: , h.t.d.
Remember: from grounds degree is rendered as reverse number, unlike the previous case!
Property 6: Derivation of the exponent from the base and the argument of the logarithm:
Or if the degrees are the same: .
Property 7: Transition to new base:
Proof: Let, then.
We have: , h.t.d.
Property 8: Swapping the base and the argument of the logarithm:
Proof: This is a special case of formula 7: if we substitute, we get: , p.t.d.
Let's look at a few more examples.
Example 4
Find the value of the expression.
We use the property of logarithms No. 2 - the sum of logarithms with the same base is equal to the logarithm of the product:
Example 5
Find the value of the expression.
Solution:
We use the property of logarithms No. 3 and No. 4:
Example 6
Find the value of the expression.
Solution:
Using property number 7 - go to base 2:
Example 7
Find the value of the expression.
Solution:
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At the Unified State Exam and OGE and in general in life
The main properties of the logarithm, the graph of the logarithm, the domain of definition, the set of values, the basic formulas, the increase and decrease are given. Finding the derivative of the logarithm is considered. As well as integral, power series expansion and representation by means of complex numbers.
ContentDomain, set of values, ascending, descending
The logarithm is a monotonic function, so it has no extremums. The main properties of the logarithm are presented in the table.
| Domain | 0 < x < + ∞ | 0 < x < + ∞ |
| Range of values | - ∞ < y < + ∞ | - ∞ < y < + ∞ |
| Monotone | increases monotonically | decreases monotonically |
| Zeros, y= 0 | x= 1 | x= 1 |
| Points of intersection with the y-axis, x = 0 | No | No |
| + ∞ | - ∞ | |
| - ∞ | + ∞ |
Private values
The base 10 logarithm is called decimal logarithm and is marked like this:
base logarithm e called natural logarithm:
Basic logarithm formulas
Properties of the logarithm following from the definition of the inverse function:
The main property of logarithms and its consequences
Base replacement formula
Logarithm is the mathematical operation of taking a logarithm. When taking a logarithm, the products of factors are converted to sums of terms.
Potentiation is the mathematical operation inverse to logarithm. When potentiating, the given base is raised to the power of the expression on which the potentiation is performed. In this case, the sums of terms are converted into products of factors.
Proof of the basic formulas for logarithms
Formulas related to logarithms follow from formulas for exponential functions and from the definition of an inverse function.
Consider the property of the exponential function
.
Then
.
Apply the property of the exponential function
:
.
Let us prove the base change formula.
;
.
Setting c = b , we have:
Inverse function
The reciprocal of the base a logarithm is the exponential function with exponent a.
If , then
If , then
Derivative of the logarithm
Derivative of logarithm modulo x :
.
Derivative of the nth order:
.
Derivation of formulas > > >
To find the derivative of a logarithm, it must be reduced to the base e.
;
.
Integral
The integral of the logarithm is calculated by integrating by parts : .
So,
Expressions in terms of complex numbers
Consider the complex number function z:
.
Let's express a complex number z via module r and argument φ
:
.
Then, using the properties of the logarithm, we have:
.
Or
However, the argument φ
not clearly defined. If we put
, where n is an integer,
then it will be the same number for different n.
Therefore, the logarithm, as a function of a complex variable, is not a single-valued function.
Power series expansion
For , the expansion takes place:
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
Let's explain it easier. For example, \(\log_(2)(8)\) is equal to the power \(2\) must be raised to to get \(8\). From this it is clear that \(\log_(2)(8)=3\).
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Examples: |
\(\log_(5)(25)=2\) |
because \(5^(2)=25\) |
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\(\log_(3)(81)=4\) |
because \(3^(4)=81\) |
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\(\log_(2)\)\(\frac(1)(32)\) \(=-5\) |
because \(2^(-5)=\)\(\frac(1)(32)\) |
Argument and base of the logarithm
Any logarithm has the following "anatomy":
The argument of the logarithm is usually written at its level, and the base is written in subscript closer to the sign of the logarithm. And this entry is read like this: "the logarithm of twenty-five to the base of five."
How to calculate the logarithm?
To calculate the logarithm, you need to answer the question: to what degree should the base be raised to get the argument?
For example, calculate the logarithm: a) \(\log_(4)(16)\) b) \(\log_(3)\)\(\frac(1)(3)\) c) \(\log_(\sqrt (5))(1)\) d) \(\log_(\sqrt(7))(\sqrt(7))\) e) \(\log_(3)(\sqrt(3))\)
a) To what power must \(4\) be raised to get \(16\)? Obviously the second. That's why:
\(\log_(4)(16)=2\)
\(\log_(3)\)\(\frac(1)(3)\) \(=-1\)
c) To what power must \(\sqrt(5)\) be raised to get \(1\)? And what degree makes any number a unit? Zero, of course!
\(\log_(\sqrt(5))(1)=0\)
d) To what power must \(\sqrt(7)\) be raised to get \(\sqrt(7)\)? In the first - any number in the first degree is equal to itself.
\(\log_(\sqrt(7))(\sqrt(7))=1\)
e) To what power must \(3\) be raised to get \(\sqrt(3)\)? From we know that is a fractional power, and therefore the square root is the power of \(\frac(1)(2)\) .
\(\log_(3)(\sqrt(3))=\)\(\frac(1)(2)\)
Example : Calculate the logarithm \(\log_(4\sqrt(2))(8)\)
Solution :
|
\(\log_(4\sqrt(2))(8)=x\) |
We need to find the value of the logarithm, let's denote it as x. Now let's use the definition of the logarithm: |
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\((4\sqrt(2))^(x)=8\) |
What links \(4\sqrt(2)\) and \(8\)? Two, because both numbers can be represented by twos: |
|
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\(((2^(2)\cdot2^(\frac(1)(2))))^(x)=2^(3)\) |
On the left, we use the degree properties: \(a^(m)\cdot a^(n)=a^(m+n)\) and \((a^(m))^(n)=a^(m\cdot n)\) |
|
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\(2^(\frac(5)(2)x)=2^(3)\) |
The bases are equal, we proceed to the equality of indicators |
|
|
\(\frac(5x)(2)\) \(=3\) |
|
Multiply both sides of the equation by \(\frac(2)(5)\) |
|
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The resulting root is the value of the logarithm |
Answer : \(\log_(4\sqrt(2))(8)=1,2\)
Why was the logarithm invented?
To understand this, let's solve the equation: \(3^(x)=9\). Just match \(x\) to make the equality work. Of course, \(x=2\).
Now solve the equation: \(3^(x)=8\). What is x equal to? That's the point.
The most ingenious will say: "X is a little less than two." How exactly is this number to be written? To answer this question, they came up with the logarithm. Thanks to him, the answer here can be written as \(x=\log_(3)(8)\).
I want to emphasize that \(\log_(3)(8)\), as well as any logarithm is just a number. Yes, it looks unusual, but it is short. Because if we wanted to write it as a decimal, it would look like this: \(1.892789260714.....\)
Example : Solve the equation \(4^(5x-4)=10\)
Solution :
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\(4^(5x-4)=10\) |
\(4^(5x-4)\) and \(10\) cannot be reduced to the same base. So here you can not do without the logarithm. Let's use the definition of the logarithm: |
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\(\log_(4)(10)=5x-4\) |
Flip the equation so x is on the left |
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\(5x-4=\log_(4)(10)\) |
Before us. Move \(4\) to the right. And don't be afraid of the logarithm, treat it like a regular number. |
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\(5x=\log_(4)(10)+4\) |
Divide the equation by 5 |
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\(x=\)\(\frac(\log_(4)(10)+4)(5)\) |
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Here is our root. Yes, it looks unusual, but the answer is not chosen. |
Answer : \(\frac(\log_(4)(10)+4)(5)\)
Decimal and natural logarithms
As stated in the definition of the logarithm, its base can be any positive number except one \((a>0, a\neq1)\). And among all the possible bases, there are two that occur so often that a special short notation was invented for logarithms with them:
Natural logarithm: a logarithm whose base is the Euler number \(e\) (equal to approximately \(2.7182818…\)), and the logarithm is written as \(\ln(a)\).
That is, \(\ln(a)\) is the same as \(\log_(e)(a)\)
Decimal logarithm: A logarithm whose base is 10 is written \(\lg(a)\).
That is, \(\lg(a)\) is the same as \(\log_(10)(a)\), where \(a\) is some number.
Basic logarithmic identity
Logarithms have many properties. One of them is called "Basic logarithmic identity" and looks like this:
| \(a^(\log_(a)(c))=c\) |
This property follows directly from the definition. Let's see how this formula came about.
Recall the short definition of the logarithm:
if \(a^(b)=c\), then \(\log_(a)(c)=b\)
That is, \(b\) is the same as \(\log_(a)(c)\). Then we can write \(\log_(a)(c)\) instead of \(b\) in the formula \(a^(b)=c\) . It turned out \(a^(\log_(a)(c))=c\) - the main logarithmic identity.
You can find the rest of the properties of logarithms. With their help, you can simplify and calculate the values of expressions with logarithms, which are difficult to calculate directly.
Example : Find the value of the expression \(36^(\log_(6)(5))\)
Solution :
Answer : \(25\)
How to write a number as a logarithm?
As mentioned above, any logarithm is just a number. The converse is also true: any number can be written as a logarithm. For example, we know that \(\log_(2)(4)\) is equal to two. Then you can write \(\log_(2)(4)\) instead of two.
But \(\log_(3)(9)\) is also equal to \(2\), so you can also write \(2=\log_(3)(9)\) . Similarly with \(\log_(5)(25)\), and with \(\log_(9)(81)\), etc. That is, it turns out
\(2=\log_(2)(4)=\log_(3)(9)=\log_(4)(16)=\log_(5)(25)=\log_(6)(36)=\ log_(7)(49)...\)
Thus, if we need, we can write the two as a logarithm with any base anywhere (even in an equation, even in an expression, even in an inequality) - we just write the squared base as an argument.
It's the same with a triple - it can be written as \(\log_(2)(8)\), or as \(\log_(3)(27)\), or as \(\log_(4)(64) \) ... Here we write the base in the cube as an argument:
\(3=\log_(2)(8)=\log_(3)(27)=\log_(4)(64)=\log_(5)(125)=\log_(6)(216)=\ log_(7)(343)...\)
And with four:
\(4=\log_(2)(16)=\log_(3)(81)=\log_(4)(256)=\log_(5)(625)=\log_(6)(1296)=\ log_(7)(2401)...\)
And with minus one:
\(-1=\) \(\log_(2)\)\(\frac(1)(2)\) \(=\) \(\log_(3)\)\(\frac(1)( 3)\) \(=\) \(\log_(4)\)\(\frac(1)(4)\) \(=\) \(\log_(5)\)\(\frac(1 )(5)\) \(=\) \(\log_(6)\)\(\frac(1)(6)\) \(=\) \(\log_(7)\)\(\frac (1)(7)\)\(...\)
And with one third:
\(\frac(1)(3)\) \(=\log_(2)(\sqrt(2))=\log_(3)(\sqrt(3))=\log_(4)(\sqrt( 4))=\log_(5)(\sqrt(5))=\log_(6)(\sqrt(6))=\log_(7)(\sqrt(7))...\)
Any number \(a\) can be represented as a logarithm with base \(b\): \(a=\log_(b)(b^(a))\)
Example : Find the value of an expression \(\frac(\log_(2)(14))(1+\log_(2)(7))\)
Solution :
Answer : \(1\)
What is a logarithm?
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
What is a logarithm? How to solve logarithms? These questions confuse many graduates. Traditionally, the topic of logarithms is considered complex, incomprehensible and scary. Especially - equations with logarithms.
This is absolutely not true. Absolutely! Don't believe? Good. Now, for some 10 - 20 minutes you:
1. Understand what is a logarithm.
2. Learn to solve a whole class of exponential equations. Even if you haven't heard of them.
3. Learn to calculate simple logarithms.
Moreover, for this you will only need to know the multiplication table, and how a number is raised to a power ...
I feel you doubt ... Well, keep time! Go!
First, solve the following equation in your mind:
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You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
