What natural logarithm is equal to 1. Natural logarithm

often take a number e = 2,718281828 . Logarithms based on 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 basis, are not indicated.

In other words, the formulation will look like: natural logarithm numbers X- this is an exponent to which a number must be raised e, To obtain x.

So, ln(7,389...)= 2, since e 2 =7,389... . Natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is zero, since e 0 = 1.

The number itself e defines the limit of a monotonic 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. Speed ​​of memorizing the first nine digits of a number e after the decimal point will increase if you notice that 1828 is the year of birth of Leo Tolstoy!

Today there are enough full tables natural logarithms.

Natural logarithm graph(functions y =ln x) is a consequence of the exponential graph mirror image relatively straight y = x and has the form:

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 is consistent with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name “natural”.

If you 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:

e ln(a) =a (a>0)

ln(e a) =a

By analogy with all logarithms, the natural logarithm converts multiplication into addition, division into 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 logarithm graph, we find that it exists for positive values variable x. It increases monotonically in 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 quite slowly. Any power function xa 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 when passing higher mathematics. Thus, using the logarithm is convenient for finding the answer to equations in which unknowns appear as exponents. The use of natural logarithms in calculations makes it possible to greatly simplify a large number of mathematical formulas. Logarithms to the base e are present when solving a significant number physical problems and naturally enter into mathematical description 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 areas of mathematics and practical sciences, they are resorted to in the field of finance to solve large number tasks, including the calculation of compound interest.

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 Furthermore where we started.
• 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 - they are 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 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 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, 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 with continuous growth, and now you are talking about annual accrual? 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 growth in interest rates, population growth, bacterial cultures, and anything 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

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 obtain 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 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 regarding multiplication by group 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 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 a table of natural logarithms back in 1619. 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 to denote the 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 derivative problem 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. So, for example, there are several simple rows 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 assuming 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 a better convergence rate, you 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 they used numeric tables and manipulations similar to those described above were performed.

High accuracy

To calculate the natural logarithm with big amount 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 a very high precision calculation 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 for the most part 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.

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

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 for positive values ​​of the 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. Basic properties 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:

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