logarithm of a given number is called the exponent to which another number must be raised, called basis logarithm to get the given number. For example, the logarithm of the number 100 to base 10 is 2. In other words, 10 must be squared to get the number 100 (10 2 = 100). If a n- a given number, b- base and l is the logarithm, then bl = n. Number n also called the base antilogarithm b numbers l. For example, the antilogarithm of 2 to base 10 is 100. This can be written as log b n = l and antilog b l = n.
The main properties of logarithms:
Any positive number other than one can be the base of logarithms, but unfortunately it turns out that if b and n are rational numbers, then in rare cases there is such a rational number l, what bl = n. However, one can define an irrational number l, for example, such that 10 l= 2; it's an irrational number l can be approximated by rational numbers with any required accuracy. It turns out that in this example l is approximately 0.3010, and this approximate value of the base 10 logarithm of 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 written as log2 = 0.3010 or log2 = 0.3010, omitting the explicit indication of the base of the logarithm. base logarithms 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 e 0,6931 = 2.
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 10 0 = 1, 10 1 = 10 and 10 2 = 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 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 between 0 and 1. The integer part called characteristic logarithm and is determined by the number itself, the fractional part is called mantissa and can be found from 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(2ё10) = 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 with 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, we note that the number 3.59 is between 10 0 and 10 1, so its characteristic is 0. We find the number 35 (on the left) in the table 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 10 2 and 10 3, 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 \u003d 10ґ54.32 \u003d 10 2 ґ5.432. Thus, by the natural logarithm of a given number a you can find the natural logarithms of numbers, equal to the products of the number a to any degree n number 10 if k ln a add ln10 multiplied by n, i.e. ln( aґ10n) = log a + n ln10 = ln a + 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 a x=ln y, then y = e x, and y called the exponent x(for the convenience of typographical typesetting, they often write y=exp x). The exponent plays the role of the antilogarithm of the number x.
With the help of tables of decimal and natural logarithms, you can create tables of logarithms in any base other than 10 and e. If log b a = x, then b x = a, and hence log c b x= log c a or x log c b= log c a, or x= log c a/log c b= log b a. Therefore, using this inversion formula from the table of logarithms to the base c you can build tables of logarithms in any other base b. Multiplier 1/log c b called transition module from the ground c to the 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 = log e 5.432/log e 10 \u003d 1.6923 / 2.3026 \u003d 1.6923´0.4343 \u003d 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 in order to use their properties log ab= log a+log b and log a/b= log a–log b, turn products into sums, and quotients into differences. In other words, if log a and log b 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, often for given values of log a and log b need to find log( a + b) or log( a – b). Of course, one could first find from tables of logarithms a and b, then perform the specified addition or subtraction and, again referring to the tables, find the required logarithms, but such a procedure would require three trips to the tables. Z. Leonelli in 1802 published the tables of the so-called. Gaussian logarithms- logarithms of addition of sums and differences - which made it possible to restrict one access 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 at a= 1 are called logarithms and are used in calculations when one has to deal with products and quotients. The logarithm of a number n equal to the logarithm of the reciprocal; those. colog n= log1/ n= -log n. 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 of the form pq/r triple sum of positive decimals log p+log q+ colog r is easier to find than the mixed sum and difference of log p+log q–log r.
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 10 8 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 Integer arithmetic(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 Stiefel's rules led J. Naper to formally introduce the first system of logarithms in the essay Description of the amazing logarithm table, 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 works in sums. The method mentioned in Napier's communication was based on the use of trigonometric formulas of the type
therefore, the Napier 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 role equivalent to the base of the system of logarithms in his system was played by the number (1 - 10 -7)ґ10 7, approximately equal to 1/ e.
Independently of Neuper and almost simultaneously with him, a system of logarithms, quite close in type, was invented and published by J. Bürgi in Prague, who published in 1620 Arithmetic and geometric progression tables. These were tables of antilogarithms in base (1 + 10 –4) ґ10 4 , a fairly good approximation of the number e.
In Napier's system, the logarithm of the number 10 7 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, the table of which Briggs published in his essay Logarithmic arithmetic(1620). base logarithms e, although not quite the ones introduced by Napier, are often referred to as Napier's. 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 began to be associated with the study of areas under a hyperbola xy= 1 (Fig. 1). In the 17th century it was shown that the area bounded by this curve, the axis x and ordinates x= 1 and x = a(in Fig. 1 this area is covered with thicker and rarer dots) increases in arithmetic progression 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=ln x, whose ordinates increase in arithmetic progression, while the abscissas increase in geometric progression, is shown in Fig. 2, a. Graph of the inverse, or exponential (exponential) function y = e x, whose ordinates increase exponentially, and the abscissas increase arithmetic, is presented, respectively, in Fig. 2, b. (Curves y= log x and y = 10x similar in shape to curves y=ln x and y = e x.) Alternative definitions of the logarithmic function have also been proposed, for example,
kpi ; and, similarly, the natural logarithms of -1 are complex numbers of the form (2 k + 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 a f(x) is a continuous function of a real number x, which has 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 by reason 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. distance from number 1 to any number x chosen equal to 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 the function satisfies a power law of the form y = kx n, then its logarithmic graph looks like a straight line, because log y= log k + n log x is an equation linear with respect to 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 = kb rx occur 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, mollusk shells twist Nautilus, horns of a mountain sheep and beaks of parrots. All of these natural shapes are examples of the curve known as the logarithmic spiral, because in polar coordinates its equation is r = ae bq, or ln r=ln a + 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 an eccentric cam and the trajectory of certain insects flying towards the light.
often take a number e = 2,718281828 . Logarithms in this base are called natural. When performing calculations with natural logarithms, it is common to operate with the sign ln, but not log; while the number 2,718281828 , defining the base, do not indicate.
In other words, the wording will look like: natural logarithm numbers X is the exponent to which the number is to be raised e, To obtain x.
So, ln(7,389...)= 2 because e 2 =7,389... . The natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is equal to zero, since e 0 = 1.
The number itself e defines the limit of a monotone bounded sequence
calculated that e = 2,7182818284... .
Quite often, in order to fix a number in memory, the digits of the required number are associated with some outstanding date. The speed of remembering the first nine digits of a number e after the decimal point will increase if you note that 1828 is the year of Leo Tolstoy's birth!
To date, there are fairly complete tables of natural logarithms.
natural log graph(functions y=ln x) is a consequence of the plot of the exponent as a mirror image with respect to the straight line y = x and looks like:
The natural logarithm can be found for every positive real number a as the area under the curve y = 1/x from 1 before a.
The elementary nature of this formulation, which fits in with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name "natural".
If we analyze natural logarithm, as a real function of a real variable, then it acts inverse function to an exponential function, which reduces to the identities:
ln(a)=a (a>0)
ln(e a)=a
By analogy with all logarithms, the natural logarithm converts multiplication to addition, division to subtraction:
ln(xy) = ln(x) + ln(y)
ln(x/y)= lnx - lny
The logarithm can be found for every positive base that is not equal to one, not just for e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm.
Having analyzed natural log graph, we get that it exists for positive values of the variable x. It monotonically increases on its domain of definition.
At x → 0 the limit of the natural logarithm is minus infinity ( -∞ ).At x → +∞ the limit of the natural logarithm is plus infinity ( + ∞ ). At large x the logarithm increases rather slowly. Any power function x a with a positive exponent a increases faster than the logarithm. The natural logarithm is a monotonically increasing function, so it has no extrema.
Usage natural logarithms very rational in the passage of higher mathematics. Thus, the use of the logarithm is convenient for finding the answer to equations in which the unknowns appear as an exponent. The use of natural logarithms in calculations makes it possible to greatly facilitate a large number of mathematical formulas. base logarithms e are present in solving a significant number of physical problems and are naturally included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the decay constant for a known half-life, or to calculate the decay time in solving problems of radioactivity. They play a leading role in many sections of mathematics and practical sciences, they are resorted to in the field of finance to solve a large number of problems, including in the calculation of compound interest.
natural logarithm
Graph of the natural logarithm function. The function slowly approaches positive infinity as x and rapidly approaches negative infinity when x tends to 0 (“slowly” and “fastly” compared to any power function of x).
natural logarithm is the base logarithm , where e is an irrational constant equal to approximately 2.718281 828 . The natural logarithm is usually denoted as ln( x), log e (x) or sometimes just log( x) if the base e implied.
Natural logarithm of a number x(written as log(x)) is the exponent to which you want to raise the number e, To obtain x. For example, ln(7,389...) equals 2 because e 2 =7,389... . The natural logarithm of the number itself e (ln(e)) is equal to 1 because e 1 = e, and the natural logarithm 1 ( log(1)) is 0 because e 0 = 1.
The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is consistent with many other formulas that use the natural logarithm, has led to the name "natural". This definition can be extended to complex numbers, which will be discussed below.
If we consider the natural logarithm as a real function of a real variable, then it is the inverse function of the exponential function, which leads to the identities:
Like all logarithms, the natural logarithm maps multiplication to addition:
Thus, the logarithmic function is an isomorphism of the group of positive real numbers with respect to multiplication by the group of real numbers by addition, which can be represented as a function:
The logarithm can be defined for any positive base other than 1, not just e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm. Logarithms are useful for solving equations in which the unknowns are present as an exponent. For example, logarithms are used to find the decay constant for a known half-life, or to find the decay time in solving problems of radioactivity. They play an important role in many areas of mathematics and applied sciences, are used in the field of finance to solve many problems, including finding compound interest.
Story
The first mention of the natural logarithm was made by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although mathematics teacher John Spydell compiled a table of natural logarithms back in 1619. Previously, it was called the hyperbolic logarithm because it corresponds to the area under the hyperbola. It is sometimes called the Napier logarithm, although the original meaning of this term was somewhat different.
Notation Conventions
The natural logarithm is usually denoted by "ln( x)”, base 10 logarithm through “lg( x)", and it is customary to indicate other grounds explicitly with the symbol "log".
In many papers on discrete mathematics, cybernetics, computer science, the authors use the notation “log( x)" for logarithms to base 2, but this convention is not universally accepted and requires clarification, either in a list of notation used or (if no such list exists) by a footnote or comment on first use.
The parentheses around the argument of logarithms (if this does not lead to an erroneous reading of the formula) are usually omitted, and when raising the logarithm to a power, the exponent is attributed directly to the sign of the logarithm: ln 2 ln 3 4 x 5 = [ ln ( 3 )] 2 .
Anglo-American system
Mathematicians, statisticians and some engineers usually use either "log( x)", or "ln( x)" , and to denote the logarithm to base 10 - "log 10 ( x)».
Some engineers, biologists, and other professionals always write "ln( x)" (or occasionally "log e ( x)") when they mean the natural logarithm, and the notation "log( x)" means log 10 ( x).
log e is the "natural" logarithm because it occurs automatically and appears very often in mathematics. For example, consider the problem of the derivative of a logarithmic function:
If the base b equals e, then the derivative is simply 1/ x, and when x= 1 this derivative is equal to 1. Another justification for which the base e logarithm is the most natural, is that it can be quite simply defined in terms of a simple integral or Taylor series, which cannot be said about other logarithms.
Further substantiations of naturalness are not connected with the number. So, for example, there are several simple series with natural logarithms. Pietro Mengoli and Nicholas Mercator called them logarithmus naturalis several decades until Newton and Leibniz developed differential and integral calculus.
Definition
Formally ln( a) can be defined as the area under the curve of the graph 1/ x from 1 to a, i.e. as an integral:
It is indeed a logarithm since it satisfies the fundamental property of a logarithm:
This can be demonstrated by assuming the following:
Numerical value
To calculate the numerical value of the natural logarithm of a number, you can use its expansion in a Taylor series in the form:
To get the best rate of convergence, you can use the following identity:
For ln( x), where x> 1, the closer the value x to 1, the faster the convergence rate. The identities associated with the logarithm can be used to achieve the goal:
These methods were used even before the advent of calculators, for which numerical tables were used and manipulations similar to those described above were performed.
High accuracy
For calculating the natural logarithm with many digits of precision, the Taylor series is not efficient because its convergence is slow. An alternative is to use Newton's method to invert into an exponential function, whose series converges faster.
An alternative for very high calculation accuracy is the formula:
where M denotes the arithmetic-geometric mean of 1 and 4/s, and
m chosen so that p marks of accuracy is achieved. (In most cases, a value of 8 for m is sufficient.) Indeed, if this method is used, Newton's inversion of the natural logarithm can be applied to efficiently calculate the exponential function. (The constants ln 2 and pi can be precomputed to the desired accuracy using any of the known rapidly convergent series.)
Computational complexity
The computational complexity of natural logarithms (using the arithmetic-geometric mean) is O( M(n)ln n). Here n is the number of digits of precision for which the natural logarithm is to be evaluated, and M(n) is the computational complexity of multiplying two n-digit numbers.
Continued fractions
Although there are no simple continued fractions to represent the logarithm, several generalized continued fractions can be used, including:
Complex logarithms
The exponential function can be extended to a function that gives a complex number of the form e x for any arbitrary complex number x, while using an infinite series with a complex x. This exponential function can be inverted to form a complex logarithm that will have most of the properties of ordinary logarithms. There are, however, two difficulties: there is no x, for which e x= 0, and it turns out that e 2pi = 1 = e 0 . Since the multiplicativity property is valid for a complex exponential function, then e z = e z+2npi for all complex z and whole n.
The logarithm cannot be defined on the entire complex plane, and even so it is multivalued - any complex logarithm can be replaced by an "equivalent" logarithm by adding any integer multiple of 2 pi. The complex logarithm can only be single-valued on a slice of the complex plane. For example ln i = 1/2 pi or 5/2 pi or −3/2 pi, etc., and although i 4 = 1.4log i can be defined as 2 pi, or 10 pi or -6 pi, etc.
see also
- John Napier - inventor of logarithms
Notes
- Mathematics for physical chemistry. - 3rd. - Academic Press, 2005. - P. 9. - ISBN 0-125-08347-5, Extract of page 9
- J J O "Connor and E F Robertson The number e . The MacTutor History of Mathematics archive (September 2001). archived
- Cajori Florian A History of Mathematics, 5th ed. - AMS Bookstore, 1991. - P. 152. - ISBN 0821821024
- Flashman, Martin Estimating Integrals using Polynomials . Archived from the original on February 12, 2012.
1.1. Determining the degree for an integer exponent
X 1 = XX 2 = X * X
X 3 = X * X * X
…
X N \u003d X * X * ... * X - N times
1.2. Zero degree.
By definition, it is customary to assume that the zero power of any number is equal to 1:1.3. negative degree.
X-N = 1/XN1.4. Fractional exponent, root.
X 1/N = N-th root of X.For example: X 1/2 = √X.
1.5. The formula for adding powers.
X (N+M) = X N * X M1.6. Formula for subtracting degrees.
X (N-M) = X N / X M1.7. Power multiplication formula.
XN*M = (XN)M1.8. The formula for raising a fraction to a power.
(X/Y)N = XN /YN2. Number e.
The value of the number e is equal to the following limit:E = lim(1+1/N), as N → ∞.
With a precision of 17 digits, the number e is 2.71828182845904512.
3. Euler's equality.
This equality links five numbers that play a special role in mathematics: 0, 1, the number e, the number pi, the imaginary unit.E(i*pi) + 1 = 0
4. Exponential function exp (x)
exp(x) = e x5. Derivative of the exponential function
An exponential function has a remarkable property: the derivative of a function is equal to the exponential function itself:(exp(x))" = exp(x)
6. Logarithm.
6.1. Definition of the logarithm function
If x = b y , then the logarithm is the functionY = Logb(x).
The logarithm shows to what degree it is necessary to raise a number - the base of the logarithm (b) to get a given number (X). The logarithm function is defined for X greater than zero.
For example: Log 10 (100) = 2.
6.2. Decimal logarithm
This is the logarithm to base 10:Y = Log 10 (x) .
Denoted Log(x): Log(x) = Log 10 (x).
An example of using the decimal logarithm is decibel.
6.3. Decibel
Item is highlighted on a separate page Decibel6.4. binary logarithm
This is the base 2 logarithm:Y = Log2(x).
Denoted by Lg(x): Lg(x) = Log 2 (X)
6.5. natural logarithm
This is the logarithm to base e:Y = loge(x) .
Denoted by Ln(x): Ln(x) = Log e (X)
The natural logarithm is the inverse of the exponential function exp(X).
6.6. characteristic points
Loga(1) = 0Log a(a) = 1
6.7. The formula for the logarithm of the product
Log a (x*y) = Log a (x)+Log a (y)6.8. The formula for the logarithm of the quotient
Log a (x/y) = Log a (x) - Log a (y)6.9. Power logarithm formula
Log a (x y) = y*Log a (x)6.10. Formula for converting to a logarithm with a different base
Log b (x) = (Log a (x)) / Log a (b)Example:
Log 2 (8) = Log 10 (8) / Log 10 (2) =
0.903089986991943552 / 0.301029995663981184 = 3
7. Formulas useful in life
Often there are problems of converting volume into area or length, and the inverse problem is converting area into volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be sheathed with boards contained in a certain volume, see the calculation of boards, how many boards are in a cube. Or, the dimensions of the wall are known, it is necessary to calculate the number of bricks, see brick calculation.
It is allowed to use the site materials provided that an active link to the source is set.
So, we have powers of two. If you take the number from the bottom line, then you can easily find the power to which you have to raise a 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 - in fact, the definition of the logarithm:
The logarithm to the base a of the argument x is the power to which the number a must be raised to get the number x .
Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.
For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6 because 2 6 = 64 .
The operation of finding the logarithm of a number to a given base is called the logarithm. So let's add a new row 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 considered 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 segment. 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 indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5 , log 3 8 , log 5 100 .
It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the base is and where the argument is. To avoid annoying misunderstandings, just take a look at the picture:
Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.
We figured out the definition - it remains 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 base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced.
- The base must be different from unity, since a unit to any power is still a unit. 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 valid range(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) is not imposed. For example, the logarithm may well be negative: log 2 0.5 \u003d -1, because 0.5 = 2 −1 .
However, now we are considering only numerical expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions that do not necessarily correspond to the above restrictions.
Now consider the general scheme for calculating logarithms. It consists of three steps:
- Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
- Solve the equation for the 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 seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.
Let's see how this scheme works with specific examples:
Task. Calculate the logarithm: log 5 25
- Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;
- Received an answer: 2.
Let's make 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 represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
- Let's make and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3 ; - Received an answer: 3.
Task. Calculate the logarithm: log 16 1
- Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
- Let's make and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0 ; - Received a response: 0.
Task. Calculate the logarithm: log 7 14
- Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;
- It follows from the previous paragraph that the logarithm is not considered;
- The answer is no change: log 7 14.
A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If there are at least two distinct factors in the expansion, the number is not an exact power.
Task. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen .
8 \u003d 2 2 2 \u003d 2 3 - the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 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 designation.
The decimal logarithm of the x argument is the base 10 logarithm, i.e. the power to which you need to raise the number 10 to get the number x. Designation: lg x .
For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.
From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x
Everything that is true for ordinary logarithms is also true for decimals.
natural logarithm
There is another logarithm that has its own notation. In a sense, it is even more important than decimal. This is the natural logarithm.
The natural logarithm of x is the base e logarithm, i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x .
Many will ask: what else is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers:
e = 2.718281828459...
We will not delve into 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 ; log 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, unity: ln 1 = 0.
For natural logarithms, all the rules that are true for ordinary logarithms are valid.
