Natural logarithm

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 of x).

Natural logarithm is the logarithm to the base , Where e- an irrational constant equal to approximately 2.718281 828. The natural logarithm is usually written as ln( x), log e (x) or sometimes just log( x), if the base e implied.

Natural logarithm of a number x(written as ln(x)) is the exponent to which the number must be raised e to get x. For example, ln(7,389...) is equal to 2 because e 2 =7,389... . Natural logarithm of the number itself e (ln(e)) is equal to 1 because e 1 = e, and the natural logarithm is 1 ( ln(1)) is equal to 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, led to the name "natural". This definition can be extended to complex numbers, as 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 with respect to 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 that involve unknowns as exponents. For example, logarithms are used to find the decay constant for a known half-life, or to find the decay time in solving radioactivity problems. They are playing important role in many areas of mathematics and applied sciences, are used in 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 the mathematics teacher John Spidell compiled the table back in 1619 natural logarithms. It was previously 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.

Designation conventions

The natural logarithm is usually denoted by “ln( x)", logarithm to base 10 - via "lg( x)", and other reasons are usually indicated explicitly with the symbol "log".

In many works on discrete mathematics, cybernetics, and computer science, authors use the notation “log( x)" for logarithms to base 2, but this convention is not generally accepted and requires clarification either in the list of notations used or (in the absence of such a list) by a footnote or comment when first used.

Parentheses around the argument of logarithms (if this does not lead to an erroneous reading of the formula) are usually omitted, and when raising a logarithm to a power, the exponent is assigned 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 the term “natural logarithm” or “log( x)" or "ln( x)", and to denote the base 10 logarithm - "log 10 ( x)».

Some engineers, biologists and other specialists always write “ln( x)" (or occasionally "log e ( x)") when they mean the natural logarithm, and the notation "log( x)" they mean log 10 ( x).

log e is a "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 reason why the base e The most natural thing about the logarithm is that it can be defined quite simply in terms of a simple integral or Taylor series, which cannot be said about other logarithms.

Further justifications for naturalness are not related to notation. 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 truly a logarithm because it satisfies the fundamental property of the logarithm:

This can be demonstrated by allowing as follows:

Numerical value

To calculate the numerical value of the natural logarithm of a number, you can use its Taylor series expansion in the form:

To get better speed convergence, we can use the following identity:

provided that y = (x−1)/(x+1) and x > 0.

For ln( x), Where x> 1, the closer the value x to 1, then faster speed convergence. 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

To calculate the natural logarithm with a large number accuracy numbers, 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 more quickly.

An alternative for very high calculation accuracy is the formula:

Where M denotes the arithmetic-geometric average 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.) In fact, if this method is used, Newton's inverse of the natural logarithm can be applied to efficiently calculate the exponential function. (The constants ln 2 and pi can be pre-calculated 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 must 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 a 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, in this case an infinite series with complex x. This exponential function can be inverted to form a complex logarithm, which 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 2πi = 1 = e 0 . Since the multiplicativity property is valid for a complex exponential function, then e z = e z+2nπi for all complex z and whole n.

The logarithm cannot be defined over 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 πi. The complex logarithm can only be single-valued on a slice of the complex plane. For example, ln i = 1/2 πi or 5/2 πi or −3/2 πi, etc., and although i 4 = 1.4 log i can be defined as 2 πi, or 10 πi or −6 πi, and so on.

See also

• John Napier - inventor of logarithms

Notes

1. Mathematics for physical chemistry. - 3rd. - Academic Press, 2005. - P. 9. - ISBN 0-125-08347-5,Extract of page 9
2. J J O"Connor and E F Robertson The number e. The MacTutor History of Mathematics archive (September 2001). Archived
3. Cajori Florian A History of Mathematics, 5th ed. - AMS Bookstore, 1991. - P. 152. - ISBN 0821821024
4. Flashman, Martin Estimating Integrals using Polynomials. Archived from the original on February 12, 2012.

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 counting

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

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 much time 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 value: 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. A negative bacteria count simply 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 receive?”

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 bet * 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, required time 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% 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?

I hope 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

As you know, when multiplying expressions with powers, their exponents always add up (a b *a c = a b+c). This mathematical law was derived by Archimedes, and later, in the 8th century, the mathematician Virasen created a table of integer exponents. It was they who served for the further discovery of logarithms. Examples of using this function can be found almost everywhere where it is necessary to simplify cumbersome multiplication by simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. In simple and accessible language.

Definition in mathematics

A logarithm is an expression of the following form: log a b=c, that is, the logarithm of any non-negative number (that is, any positive) “b” to its base “a” is considered to be the power “c” to which it is necessary to raise the base “a” in order to ultimately get the value "b". Let's analyze the logarithm using examples, let's say there is an expression log 2 8. How to find the answer? It’s very simple, you need to find a power such that from 2 to the required power you get 8. After doing some calculations in your head, we get the number 3! And that’s true, because 2 to the power of 3 gives the answer as 8.

Types of logarithms

For many pupils and students, this topic seems complicated and incomprehensible, but in fact logarithms are not so scary, the main thing is to understand their general meaning and remember their properties and some rules. There are three individual species logarithmic expressions:

1. Natural logarithm ln a, where the base is the Euler number (e = 2.7).
2. Decimal a, where the base is 10.
3. Logarithm of any number b to base a>1.

Each of them is solved in a standard way, including simplification, reduction and subsequent reduction to a single logarithm using logarithmic theorems. To receive correct values logarithms, you should remember their properties and the sequence of actions when solving them.

Rules and some restrictions

In mathematics, there are several rules-constraints that are accepted as an axiom, that is, they are not subject to discussion and are the truth. For example, it is impossible to divide numbers by zero, and it is also impossible to extract an even root from negative numbers. Logarithms also have their own rules, following which you can easily learn to work even with long and capacious logarithmic expressions:

• The base “a” must always be greater than zero, and not equal to 1, otherwise the expression will lose its meaning, because “1” and “0” to any degree are always equal to their values;
• if a > 0, then a b >0, it turns out that “c” must also be greater than zero.

How to solve logarithms?

For example, the task is given to find the answer to the equation 10 x = 100. This is very easy, you need to choose a power by raising the number ten to which we get 100. This, of course, is 10 2 = 100.

Now let's imagine this expression in logarithmic form. We get log 10 100 = 2. When solving logarithms, all actions practically converge to find the power to which it is necessary to enter the base of the logarithm in order to obtain a given number.

To accurately determine the value of an unknown degree, you need to learn how to work with a table of degrees. It looks like this:

As you can see, some exponents can be guessed intuitively if you have a technical mind and knowledge of the multiplication table. However, for larger values ​​you will need a power table. It can be used even by those who know nothing at all about complex mathematical topics. The left column contains numbers (base a), the top row of numbers is the value of the power c to which the number a is raised. At the intersection, the cells contain the number values ​​that are the answer (a c =b). Let's take, for example, the very first cell with the number 10 and square it, we get the value 100, which is indicated at the intersection of our two cells. Everything is so simple and easy that even the most true humanist will understand!

Equations and inequalities

It turns out that under certain conditions the exponent is the logarithm. Therefore, any mathematical numerical expressions can be written as a logarithmic equality. For example, 3 4 =81 can be written as the base 3 logarithm of 81 equal to four (log 3 81 = 4). For negative powers the rules are the same: 2 -5 = 1/32 we write it as a logarithm, we get log 2 (1/32) = -5. One of the most fascinating sections of mathematics is the topic of “logarithms”. We will look at examples and solutions of equations below, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.

Given an expression of the following form: log 2 (x-1) > 3 - it is logarithmic inequality, since the unknown value "x" is under the sign of the logarithm. And also in the expression two quantities are compared: the logarithm of the desired number to base two is greater than the number three.

The most important difference between logarithmic equations and inequalities is that equations with logarithms (example - logarithm 2 x = √9) imply one or more specific answers numerical values, while when solving the inequality, both the range of permissible values ​​and the breakpoints of this function are determined. As a consequence, the answer is not a simple set of individual numbers, as in the answer to an equation, but a continuous series or set of numbers.

Basic theorems about logarithms

When solving primitive tasks of finding the values ​​of the logarithm, its properties may not be known. However, when it comes to logarithmic equations or inequalities, first of all, it is necessary to clearly understand and apply in practice all the basic properties of logarithms. We will look at examples of equations later; let's first look at each property in more detail.

1. The main identity looks like this: a logaB =B. It applies only when a is greater than 0, not equal to one, and B is greater than zero.
2. The logarithm of the product can be represented in the following formula: log d (s 1 * s 2) = log d s 1 + log d s 2. In this case prerequisite is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this logarithmic formula, with examples and solution. Let log a s 1 = f 1 and log a s 2 = f 2, then a f1 = s 1, a f2 = s 2. We obtain that s 1 * s 2 = a f1 *a f2 = a f1+f2 (properties of degrees ), and then by definition: log a (s 1 * s 2) = f 1 + f 2 = log a s1 + log a s 2, which is what needed to be proven.
3. The logarithm of the quotient looks like this: log a (s 1/ s 2) = log a s 1 - log a s 2.
4. The theorem in the form of a formula takes on next view: log a q b n = n/q log a b.

This formula is called the “property of the degree of logarithm.” It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics is based on natural postulates. Let's look at the proof.

Let log a b = t, it turns out a t =b. If we raise both parts to the power m: a tn = b n ;

but since a tn = (a q) nt/q = b n, therefore log a q b n = (n*t)/t, then log a q b n = n/q log a b. The theorem has been proven.

Examples of problems and inequalities

The most common types of problems on logarithms are examples of equations and inequalities. They are found in almost all problem books, and are also a required part of mathematics exams. For admission to university or passing entrance examinations in mathematics you need to know how to solve such problems correctly.

Unfortunately, there is no single plan or scheme for solving and determining the unknown value of the logarithm, however, it can be applied to every mathematical inequality or logarithmic equation certain rules. First of all, you should find out whether the expression can be simplified or lead to general appearance. You can simplify long logarithmic expressions if you use their properties correctly. Let's get to know them quickly.

When solving logarithmic equations, we must determine what type of logarithm we have: an example expression may contain a natural logarithm or a decimal one.

Here are examples ln100, ln1026. Their solution boils down to the fact that they need to determine the power to which the base 10 will be equal to 100 and 1026, respectively. For solutions of natural logarithms, you need to apply logarithmic identities or their properties. Let's look at examples of solving logarithmic problems of various types.

How to Use Logarithm Formulas: With Examples and Solutions

So, let's look at examples of using the basic theorems about logarithms.

1. The property of the logarithm of a product can be used in tasks where it is necessary to expand great value numbers b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4*128) = log 2 512. The answer is 9.
2. log 4 8 = log 2 2 2 3 = 3/2 log 2 2 = 1.5 - as you can see, using the fourth property of the logarithm power, we managed to solve a seemingly complex and unsolvable expression. You just need to factor the base and then take the exponent values ​​out of the sign of the logarithm.

Assignments from the Unified State Exam

Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the Unified State Exam (state exam for all school graduates). Typically, these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most complex and voluminous tasks). The exam requires accurate and perfect knowledge of the topic “Natural logarithms”.

Examples and solutions to problems are taken from official Unified State Exam options. Let's see how such tasks are solved.

Given log 2 (2x-1) = 4. Solution:
let's rewrite the expression, simplifying it a little log 2 (2x-1) = 2 2, by the definition of the logarithm we get that 2x-1 = 2 4, therefore 2x = 17; x = 8.5.

• It is best to reduce all logarithms to the same base so that the solution is not cumbersome and confusing.
• All expressions under the logarithm sign are indicated as positive, therefore, when the exponent of an expression that is under the logarithm sign and as its base is taken out as a multiplier, the expression remaining under the logarithm must be positive.

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 on 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 mirror image 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.

At x → 0 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 and argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
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:

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

So, we have powers of two. If you take the number from the bottom line, you can easily find the power to which you will have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.

And now - actually, the definition of the logarithm:

The base a logarithm of x is the power to which a must be raised to get x.

Designation: log a x = b, where a is the base, x is the argument, b is what the logarithm is actually equal to.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). With the same success log 2 64 = 6, since 2 6 = 64.

The operation of finding the logarithm of a number to a given base is called logarithmization. So, let's add a new line to our table:

2 1	2 2	2 3	2 4	2 5	2 6
2	4	8	16	32	64
log 2 2 = 1	log 2 4 = 2	log 2 8 = 3	log 2 16 = 4	log 2 32 = 5	log 2 64 = 6

Unfortunately, not all logarithms are calculated so easily. For example, try finding 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 ad infinitum, and they are never repeated. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100.

It is important to understand that a logarithm is an expression with two variables (the base and the argument). At first, many people confuse where the basis is and where the argument is. To avoid annoying misunderstandings, just look at the picture:

Before us is nothing more than the definition of a logarithm. Remember: logarithm is a power, into which the base must be built in order to obtain an argument. It is the base that is raised to a power - it is highlighted in red in the picture. It turns out that the base is always at the bottom! I tell my students this wonderful rule at the very first lesson - and no confusion arises.

We've figured out the definition - all that remains is to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:

1. The argument and the base must always be greater than zero. This follows from the definition of the degree rational indicator, to which the definition of a logarithm comes down.
2. The base must be different from one, since one to any degree still remains one. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!

Such restrictions are called range of acceptable values(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.

Note that there are no restrictions on the number b (the value of the logarithm). For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1.

However, now we are only considering numeric expressions, where it is not required to know the logarithm's CVD. All restrictions have already been taken into account by the authors of the problems. But when they go logarithmic equations and inequalities, DHS requirements will become mandatory. After all, the basis and argument may contain very strong constructions that do not necessarily correspond to the above restrictions.

Now let's consider general scheme calculating logarithms. It consists of three steps:

1. Express the base a and the argument x as a power with the minimum possible base greater than one. Along the way, it’s better to get rid of decimals;
2. Solve the equation for variable b: x = a b ;
3. The resulting number b will be the answer.

That's it! If the logarithm turns out to be irrational, this will be visible already in the first step. The requirement that the base be greater than one is very important: this reduces the likelihood of error and greatly simplifies the calculations. Same with decimals: if you immediately convert them to regular ones, there will be many fewer errors.

Let's see how this scheme works using specific examples:

Task. Calculate the logarithm: log 5 25

1. Let's imagine the base and argument as a power of five: 5 = 5 1 ; 25 = 5 2 ;

Let's create and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2 ;

2. We received the answer: 2.

Task. Calculate the logarithm:

Task. Calculate the logarithm: log 4 64

1. Let's imagine the base and argument as a power of two: 4 = 2 2 ; 64 = 2 6 ;
2. Let's create and solve the equation:
   log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3 ;
3. We received the answer: 3.

Task. Calculate the logarithm: log 16 1

1. Let's imagine the base and argument as a power of two: 16 = 2 4 ; 1 = 2 0 ;
2. Let's create and solve the equation:
   log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0 ;
3. We received the answer: 0.

Task. Calculate the logarithm: log 7 14

1. Let's imagine the base and argument as a power of seven: 7 = 7 1 ; 14 cannot be represented as a power of seven, since 7 1< 14 < 7 2 ;
2. From the previous paragraph it follows that the logarithm does not count;
3. The answer is no change: log 7 14.

A small note on the last example. How can you be sure that a number is not an exact power of another number? It’s very simple - just factor it into prime factors. If the expansion has at least two different factors, the number is not an exact power.

Task. Find out whether the numbers are exact powers: 8; 48; 81; 35; 14.

8 = 2 · 2 · 2 = 2 3 - exact degree, because there is only one multiplier;
48 = 6 · 8 = 3 · 2 · 2 · 2 · 2 = 3 · 2 4 - is not an exact power, since there are two factors: 3 and 2;
81 = 9 · 9 = 3 · 3 · 3 · 3 = 3 4 - exact degree;
35 = 7 · 5 - again not an exact power;
14 = 7 · 2 - again not an exact degree;

Let us also note that we ourselves prime numbers are always exact degrees of themselves.

Decimal logarithm

Some logarithms are so common that they have a special name and symbol.

The decimal logarithm of x is the logarithm to base 10, i.e. The power to which the number 10 must be raised to obtain the number x. Designation: lg x.

For example, log 10 = 1; lg 100 = 2; lg 1000 = 3 - etc.

From now on, when a phrase like “Find lg 0.01” appears in a textbook, know that this is not a typo. This is a decimal logarithm. However, if you are unfamiliar with this notation, you can always rewrite it:
log x = log 10 x

Everything that is true for ordinary logarithms is also true for decimal logarithms.

Natural logarithm

There is another logarithm that has its own designation. In some ways, it's even more important than decimal. It's about about the natural logarithm.

The natural logarithm of x is the logarithm to base e, i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x .

Many will ask: what is the number e? This irrational number, its exact value is impossible to find and write down. I will give only the first figures:
e = 2.718281828459...

We will not go into detail about what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x

Thus ln e = 1 ; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, for one: ln 1 = 0.

For natural logarithms, all the rules that are true for ordinary logarithms are valid.