Calculate ln from the number. LN and LOG functions for calculating the natural logarithm in EXCEL

The basic properties of the natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, power series expansion and representation of the function ln x using complex numbers are given.

Definition

Natural logarithm is the function y = ln x, the inverse of the exponential, x = e y, and is the logarithm to the base of the number e: ln x = log e x.

The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.

Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.

Graph of the function y = ln x.

Graph of natural logarithm (functions y = ln x) is obtained from the exponential graph by mirror reflection relative to the straight line y = x.

The natural logarithm is defined at positive values variable x.

It increases monotonically in its domain of definition. 0 At x →

the limit of the natural logarithm is minus infinity (-∞).

As x → + ∞, the limit of the natural logarithm is plus infinity (+ ∞). For large x, the logarithm increases quite slowly. Any power function x a with a positive exponent a grows faster than the logarithm.

Properties of the natural logarithm

Domain of definition, set of values, extrema, increase, decrease

The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.

ln x values

ln 1 = 0

Basic formulas for natural logarithms

Formulas following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Any logarithm can be expressed in terms of natural logarithms using the base substitution formula:

Proofs of these formulas are presented in the section "Logarithm".

Inverse function

The inverse of the natural logarithm is the exponent.

If , then

If, then.

Derivative ln x
.
Derivative of the natural logarithm:
.
Derivative of the natural logarithm of modulus x:
.
Derivative of nth order:

Deriving formulas > > >

Integral
.
The integral is calculated by integration by parts:

So,

Expressions using complex numbers
.
Consider the function of the complex variable z: Let's express the complex variable z via module r φ :
.
and argument
.
Using the properties of the logarithm, we have:
.
The argument φ is not uniquely defined. If you put
, where n is an integer,
it will be the same number for different n.

Therefore, the natural logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Not bad at all, right? While mathematicians search for words to give you a long, confusing definition, let's take a closer look at this simple and clear one.

The number e means growth

The number e means continuous growth. As we saw in the previous example, e x allows us to link interest and time: 3 years at 100% growth is the same as 1 year at 300%, assuming "compound interest".

You can substitute any percentage and time values (50% for 4 years), but it is better to set the percentage as 100% for convenience (it turns out 100% for 2 years). By moving to 100%, we can focus solely on the time component:

e x = e percent * time = e 1.0 * time = e time

Obviously e x means:

how much will my contribution grow after x units of time (assuming 100% continuous growth).
for example, after 3 time intervals I will receive e 3 = 20.08 times more “things”.

e x is a scaling factor that shows what level we will grow to in x amount of time.

Natural logarithm means time

The natural logarithm is the inverse of e, a fancy term for opposite. Speaking of quirks; in Latin it is called logarithmus naturali, hence the abbreviation ln.

And what does this inversion or opposite mean?

e x allows us to substitute time and get growth.
ln(x) allows us to take growth or income and find out the time it takes to generate it.

For example:

e 3 equals 20.08. After three periods of time, we will have 20.08 times more than what we started with.
ln(08/20) would be approximately 3. If you are interested in growth of 20.08 times, you will need 3 time periods (again, assuming 100% continuous growth).

Still reading? The natural logarithm shows the time required to reach the desired level.

This non-standard logarithmic count

Have you gone through logarithms? strange creatures. How did they manage to turn multiplication into addition? What about division into subtraction? Let's get a look.

What is ln(1) equal to? Intuitively, the question is: how long should I wait to get 1x more than what I have?

Zero. Zero. Not at all. You already have it once. It doesn't take long to go from level 1 to level 1.

ln(1) = 0

Okay, what about the fractional value? How long will it take for us to have 1/2 of the available quantity left? We know that with 100% continuous growth, ln(2) means the time it takes to double. If we let's turn back time(i.e., wait a negative amount of time), then we will get half of what we have.

ln(1/2) = -ln(2) = -0.693

Logical, right? If we go back (time back) to 0.693 seconds, we will find half the amount available. In general, you can turn the fraction over and take negative meaning: ln(1/3) = -ln(3) = -1.09. This means that if we go back in time to 1.09 times, we will only find a third of the current number.

Okay, what about the logarithm of a negative number? How long does it take to “grow” a colony of bacteria from 1 to -3?

This is impossible! You can't get a negative bacteria count, can you? You can get a maximum (er...minimum) of zero, but there's no way you can get a negative number from these little critters. IN negative number bacteria just doesn't make sense.

ln(negative number) = undefined

"Undefined" means that there is no amount of time that would have to wait to get a negative value.

Logarithmic multiplication is just hilarious

How long will it take to grow fourfold? Of course, you can just take ln(4). But this is too simple, we will go the other way.

You can think of quadruple growth as doubling (requiring ln(2) units of time) and then doubling again (requiring another ln(2) units of time):

Time to grow 4 times = ln(4) = Time to double and then double again = ln(2) + ln(2)

Interesting. Any growth rate, say 20, can be considered a doubling right after a 10x increase. Or growth by 4 times, and then by 5 times. Or tripling and then increasing by 6.666 times. See the pattern?

ln(a*b) = ln(a) + ln(b)

The logarithm of A times B is log(A) + log(B). This relationship immediately makes sense when viewed in terms of growth.

If you are interested in 30x growth, you can wait ln(30) in one sitting, or wait ln(3) for tripling, and then another ln(10) for 10x. The end result is the same, so of course the time must remain constant (and it does).

What about division? Specifically, ln(5/3) means: how long will it take to grow 5 times and then get 1/3 of that?

Great, growth by 5 times is ln(5). An increase of 1/3 times will take -ln(3) units of time. So,

ln(5/3) = ln(5) – ln(3)

This means: let it grow 5 times, and then “go back in time” to the point where only a third of that amount remains, so you get 5/3 growth. In general it turns out

ln(a/b) = ln(a) – ln(b)

I hope that the strange arithmetic of logarithms is starting to make sense to you: multiplying growth rates becomes adding growth time units, and dividing becomes subtracting time units. No need to memorize the rules, try to understand them.

Using the natural logarithm for arbitrary growth

Well, of course,” you say, “this is all good if the growth is 100%, but what about the 5% that I get?”

No problem. The "time" we calculate with ln() is actually a combination of interest rate and time, the same X from the e x equation. We just decided to set the percentage to 100% for simplicity, but we are free to use any numbers.

Let's say we want to achieve 30x growth: take ln(30) and get 3.4 This means:

e x = height
e 3.4 = 30

Obviously, this equation means "100% return over 3.4 years gives 30x growth." We can write this equation as follows:

e x = e rate*time
e 100% * 3.4 years = 30

We can change the values of “bet” and “time”, as long as the rate * time remains 3.4. For example, if we are interested in 30x growth, how long will we have to wait at an interest rate of 5%?

ln(30) = 3.4
rate * time = 3.4
0.05 * time = 3.4
time = 3.4 / 0.05 = 68 years

I reason like this: "ln(30) = 3.4, so at 100% growth it will take 3.4 years. If I double the growth rate, the time required will be halved."

100% for 3.4 years = 1.0 * 3.4 = 3.4
200% in 1.7 years = 2.0 * 1.7 = 3.4
50% for 6.8 years = 0.5 * 6.8 = 3.4
5% over 68 years = .05 * 68 = 3.4.

Great, right? The natural logarithm can be used with any interest rate and time because their product remains constant. You can move variable values as much as you like.

Cool example: Rule of seventy-two

The Rule of Seventy-Two is a mathematical technique that allows you to estimate how long it will take for your money to double. Now we will deduce it (yes!), and moreover, we will try to understand its essence.

How long will it take to double your money at 100% interest compounded annually?

Oops. We used the natural logarithm for the case of continuous growth, and now you are talking about annual compounding? Wouldn't this formula become unsuitable for such a case? Yes, it will, but for real interest rates like 5%, 6% or even 15%, the difference between annual compounding and continuous growth will be small. So the rough estimate works, um, roughly, so we'll pretend that we have a completely continuous accrual.

Now the question is simple: How quickly can you double with 100% growth? ln(2) = 0.693. It takes 0.693 units of time (years in our case) to double our amount with a continuous increase of 100%.

So, what if the interest rate is not 100%, but say 5% or 10%?

Easily! Since bet * time = 0.693, we will double the amount:

rate * time = 0.693
time = 0.693 / bet

It turns out that if the growth is 10%, it will take 0.693 / 0.10 = 6.93 years to double.

To simplify the calculations, let's multiply both sides by 100, then we can say "10" rather than "0.10":

time to double = 69.3 / bet, where the bet is expressed as a percentage.

Now it’s time to double at a rate of 5%, 69.3 / 5 = 13.86 years. However, 69.3 is not the most convenient dividend. Let's choose a close number, 72, which is convenient to divide by 2, 3, 4, 6, 8 and other numbers.

time to double = 72 / bet

which is the rule of seventy-two. Everything is covered.

If you need to find the time to triple, you can use ln(3) ~ 109.8 and get

time to triple = 110 / bet

What is another useful rule. The "Rule of 72" applies to height interest rates, population growth, bacterial cultures, and everything that grows exponentially.

What's next?

Hopefully the natural logarithm now makes sense to you - it shows the time it takes for any number to grow exponentially. I think it is called natural because e is a universal measure of growth, so ln can be considered in a universal way determining how long it takes to grow.

Every time you see ln(x), remember "the time it takes to grow X times". In an upcoming article I will describe e and ln in conjunction so that the fresh scent of mathematics will fill the air.

Addendum: Natural logarithm of e

Quick quiz: what is ln(e)?

a math robot will say: since they are defined as the inverse of one another, it is obvious that ln(e) = 1.
understanding person: ln(e) is the number of times it takes to grow "e" times (about 2.718). However, the number e itself is a measure of growth by a factor of 1, so ln(e) = 1.

Think clearly.

September 9, 2013

Logarithm of a given number is called the exponent to which another number must be raised, called basis logarithm to get this number. For example, the base 10 logarithm of 100 is 2. In other words, 10 must be squared to get 100 (10 2 = 100). If n– a given number, b– base and l– logarithm, then b l = n. Number n also called base antilogarithm b numbers l. For example, the antilogarithm of 2 to base 10 is equal to 100. This can be written in the form of the relations log b n = l and antilog b l = n.

Basic properties of logarithms:

Any positive number, except for unity, can serve as the basis 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 b l = n. However, it is possible to define an irrational number l, for example, such that 10 l= 2; this is an irrational number l can be approximated with any required accuracy rational numbers. It turns out that in the given example l is approximately equal to 0.3010, and this approximation of the base 10 logarithm of 2 can be found in four-digit tables decimal logarithms. Base 10 logarithms (or base 10 logarithms) are so commonly used in calculations that they are called ordinary logarithms and written as log2 = 0.3010 or log2 = 0.3010, omitting the explicit indication of the logarithm base. 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 e 0,6931 = 2.

Using tables of ordinary logarithms.

The regular logarithm of a number is an exponent to which 10 must be raised to obtain the given number. Since 10 0 = 1, 10 1 = 10 and 10 2 = 100, we immediately get that log1 = 0, log10 = 1, log100 = 2, etc. for increasing integer powers 10. Likewise, 10 –1 = 0.1, 10 –2 = 0.01 and therefore log0.1 = –1, log0.01 = –2, etc. for all integers negative powers 10. The usual logarithms of the remaining numbers are contained between the logarithms of the nearest integer powers of the number 10; log2 must be between 0 and 1, log20 must be between 1 and 2, and log0.2 must be between -1 and 0. Thus, the logarithm consists of two parts, an integer and decimal, enclosed between 0 and 1. The integer part is 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(2ґ10) = log2 + log10 = (log2) + 1. The logarithm of 2 is 0.3010, so log20 = 0.3010 + 1 = 1.3010. Similarly, log0.2 = log(2о10) = log2 – log10 = (log2) – 1 = 0.3010 – 1. After subtraction, we get log0.2 = – 0.6990. However, it is more convenient to represent log0.2 as 0.3010 – 1 or as 9.3010 – 10; can be formulated and general rule: all numbers obtained from a given number by multiplication by a power of 10 have the same mantissa, equal to the mantissa of the given number. Most tables show the mantissas of numbers in the range from 1 to 10, 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. The easiest way to learn how to use such tables is with examples. To find log3.59, first of all, we note that the number 3.59 is contained 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 at the 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 figures, it is necessary to resort to interpolation. In some tables, interpolation is facilitated by the proportions given in the last nine columns on the right side of each page of the tables. Let us now find log736.4; the number 736.4 lies between 10 2 and 10 3, therefore the characteristic of its logarithm is 2. In the table we find a row to the left of which there is 73 and column 6. At the intersection of this row and this column there is the number 8669. Among the linear parts we find column 4 . At the intersection of row 73 and column 4 there is the number 2. By adding 2 to 8669, we get the mantissa - it is equal to 8671. Thus, log736.4 = 2.8671.

Natural logarithms.

Tables and Properties natural logarithms are similar to the tables and properties of ordinary logarithms. The main difference between both 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 equal to 1.6923, respectively; 3.9949 and 6.2975. The relationship between these logarithms will become obvious if we consider the differences between them: log543.2 – log54.32 = 6.2975 – 3.9949 = 2.3026; last number is nothing more than 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 = 10ґ54.32 = 10 2ґ5.432. Thus, by the natural logarithm of a given number a you can find natural logarithms of numbers, equal to the products numbers a for any degree n numbers 10 if to ln a add ln10 multiplied by n, i.e. ln( aґ10n) = log a + n ln10 = ln a + 2,3026n. For example, ln0.005432 = ln(5.432ґ10 –3) = ln5.432 – 3ln10 = 1.6923 – (3ґ2.3026) = – 5.2155. Therefore, tables of natural logarithms, like tables of ordinary logarithms, usually contain only logarithms of numbers from 1 to 10. In the system of natural logarithms, one can talk about antilogarithms, but more often they talk about exponential function or about the exhibitor. If x= log y, That y = e x, And y called the exponent of x(for typographic convenience, they often write y= exp x). 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 log b a = x, That b x = a, and therefore 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 base logarithm table c you can build tables of logarithms in any other base b. Multiplier 1/log c b called transition module from the base c to the base b. Nothing prevents, for example, using the inversion formula or transition from one system of logarithms to another, finding natural logarithms from the table of ordinary logarithms or making the reverse transition. For example, log105.432 = log e 5.432/log e 10 = 1.6923/2.3026 = 1.6923ґ0.4343 = 0.7350. The number 0.4343, by which the natural logarithm of a given number must be multiplied to obtain an ordinary logarithm, is the modulus of the transition to the system of ordinary logarithms.

Special tables.

Logarithms were originally invented so that, using 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 using addition and subtraction we can easily find the logarithm of the product and the quotient. In astronomy, however, often 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 indicated addition or subtraction and, again referring to the tables, find the required logarithms, but such a procedure would require referring to the tables three times. Z. Leonelli in 1802 published tables of the so-called. Gaussian logarithms– logarithms for adding sums and differences – which made it possible to limit oneself to one access to tables.

In 1624, I. Kepler proposed tables of proportional logarithms, i.e. logarithms of numbers a/x, Where a– some positive constant value. These tables are used primarily by astronomers and navigators.

Proportional logarithms at a= 1 are called by logarithms and are used in calculations when one has to deal with products and quotients. Cologarithm of a number n equal to the logarithm reciprocal number; those. colog n= log1/ n= – log n. If log2 = 0.3010, then colog2 = – 0.3010 = 0.6990 – 1. The advantage of using cologarithms is that when calculating the value of the logarithm of expressions like pq/via module triple sum of positive decimals log p+ log q+colog via module is easier to find than the mixed sum and difference log p+ log q–log via module.

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 table values of positive integer powers of integers was used to calculate compound interest. Much later, Archimedes (287–212 BC) used powers of 108 to find an upper limit on the number of grains of sand required to completely fill the then known Universe. Archimedes drew attention to the property of exponents that underlies the effectiveness of logarithms: the product of powers corresponds to the sum of the exponents. At the end of the Middle Ages and the beginning of the modern era, mathematicians increasingly began to turn 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 exponent row) is equal to the exponent of two corresponding to the product of the two corresponding numbers in the bottom row (the exponent row). In connection with this table, Stiefel formulated four rules equivalent to the four modern rules for operations on exponents or the four rules for operations on logarithms: the sum on the top line corresponds to the product on the bottom line; subtraction on the top line corresponds to division on the bottom line; multiplication on the top line corresponds to exponentiation on the bottom line; division on the top line corresponds to rooting on the bottom line.

Apparently, rules similar to Stiefel’s rules led J. Naper to formally introduce the first system of logarithms in his work Description of the amazing table of logarithms, published in 1614. But Napier’s thoughts were occupied with the problem of converting products into sums ever since, more than ten years before the publication of his work, Napier received news from Denmark that at the Tycho Brahe Observatory his assistants had a method that made it possible to convert products into sums. The method mentioned in the message Napier received was based on the use trigonometric formulas type

therefore Naper's tables consisted mainly of logarithms 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 Naper and almost simultaneously with him, a system of logarithms, quite similar in type, was invented and published by J. Bürgi in Prague, published in 1620 Arithmetic and geometric progression tables. These were tables of antilogarithms to the base (1 + 10 –4) ґ10 4, a fairly good approximation of the number e.

In the Naper system, the logarithm of the number 10 7 was taken to be 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 to be zero. Then, as the numbers increased, their logarithms would increase. So we got modern system 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 Naper, are often called Naper's. The terms "characteristic" and "mantissa" were proposed by Briggs.

First logarithms in force 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 sparse dots) increases in arithmetic progression, When a increases exponentially. It is precisely this dependence that arises in the rules for operations with exponents and logarithms. This gave rise to calling Naperian 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 thanks to the work of Euler, the concept of a logarithmic function was formed. Graph of such a function y= log x, whose ordinates increase in an arithmetic progression, while the abscissas increase in a geometric progression, is presented in Fig. 2, A. Graph of an inverse or exponential function y = e x, whose ordinates increase in geometric progression, and whose abscissas increase in arithmetic progression, is presented, respectively, in Fig. 2, b. (Curves y=log x And y = 10x similar in shape to curves y= log x And y = e x.) Alternative definitions of the logarithmic function have also been proposed, e.g.

kpi ; and, similarly, the natural logarithms of the number -1 are complex numbers of the form (2 k + 1)pi, Where k– an integer. Similar statements are true for general logarithms or other systems of logarithms. Additionally, the definition of logarithms can be generalized using Euler's identities to include complex logarithms of complex numbers.

An alternative definition of a logarithmic function is provided by functional analysis. If f(x) – continuous function of a real number x, having the following three properties: f (1) = 0, f (b) = 1, f (uv) = f (u) + f (v), That f(x) is defined as the logarithm of the number x based on 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 so-called. slide rule - a computational tool whose operating principle 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 to be 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 directly from the scale the products or quotients of the corresponding numbers. You can also take advantage of the advantages of representing numbers in logarithmic form. logarithmic paper for plotting graphs (paper with logarithmic scales printed on it on 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– equation linear with respect to log y and log x. On the contrary, if the logarithmic graph of some functional dependence looks like a straight line, then this dependence is a power one. Semi-log paper (where the y-axis has a logarithmic scale and the x-axis has a uniform scale) is useful when you need to identify exponential functions. Equations of the form y = kb rx occur whenever a quantity, such as a population, a quantity of radioactive material, or a bank balance, decreases or increases at a rate proportional to the available this moment number of residents, radioactive substance or money. If such a dependence is plotted on semi-logarithmic paper, the graph will look like a straight line.

The logarithmic function arises in connection with a wide variety of natural forms. Flowers in sunflower inflorescences are arranged in logarithmic spirals, mollusk shells are twisted Nautilus, horns mountain sheep and parrot beaks. All of these natural shapes can serve as examples of a curve known as a logarithmic spiral because, in a polar coordinate system, its equation is r = ae bq, or ln via module= log a + bq. Such a curve is described by a moving point, the distance from the pole of which increases in geometric progression, and the angle described by its radius vector increases in arithmetic progression. The ubiquity of such a curve, and therefore of the logarithmic function, is well illustrated by the fact that it occurs in such distant and completely different areas as the contour of an eccentric cam and the trajectory of some insects flying towards the light.

Graph of the natural logarithm function. The function slowly approaches positive infinity as it increases x and quickly approaches negative infinity when x tends to 0 (“slow” and “fast” compared to any power function from x).

Natural logarithm is the logarithm to the base , Where e (\displaystyle e)- an irrational constant equal to approximately 2.72. It is denoted as ln ⁡ x (\displaystyle \ln x), log e ⁡ x (\displaystyle \log _(e)x) or sometimes just log ⁡ x (\displaystyle \log x), if the base e (\displaystyle e) implied . In other words, the natural logarithm of a number x- this is an exponent to which a number must be raised e, To obtain x. This definition can be extended to complex numbers.

ln ⁡ e = 1 (\displaystyle \ln e=1), because e 1 = e (\displaystyle e^(1)=e); ln ⁡ 1 = 0 (\displaystyle \ln 1=0), because e 0 = 1 (\displaystyle e^(0)=1).

The natural logarithm can also be defined geometrically for any positive real number a as the area under the curve y = 1 x (\displaystyle y=(\frac (1)(x))) in between [ 1 ; a ] (\displaystyle ). The simplicity of this definition, which is consistent with many other formulas that use this logarithm, explains the origin of the name "natural".

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:

e ln ⁡ a = a (a > 0) ; (\displaystyle e^(\ln a)=a\quad (a>0);) ln ⁡ e a = a (a > 0) . (\displaystyle \ln e^(a)=a\quad (a>0).)

Like all logarithms, the natural logarithm maps multiplication to addition:

ln ⁡ x y = ln ⁡ x + ln ⁡ y . (\displaystyle \ln xy=\ln x+\ln y.)

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

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

Basic logarithmic identity

a log a b = b (a > 0, a ≠ 1) (2)

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

Two obvious consequences of the definition of logarithm

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

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

Logarithm of the product and logarithm of the quotient

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

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

I would like to warn schoolchildren against thoughtlessly using these formulas when solving logarithmic equations and inequalities. When using them “from left to right,” the ODZ narrows, and when moving from the sum or difference of logarithms to the logarithm of the product or quotient, the ODZ expands.

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

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

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

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

And again I would like to urge caution. Consider the following example:

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

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

Formula for moving to a new foundation

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

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

If we choose the number b as the new base c, we get an important special case formulas (8):

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

Some simple examples with logarithms

Example 1. Calculate: log2 + log50.
Solution. log2 + log50 = log100 = 2. We used the sum of logarithms formula (5) and the definition of the decimal logarithm.

Example 2. Calculate: lg125/lg5.
Solution. log125/log5 = log 5 125 = 3. We used the formula for moving to a new base (8).