Natural logarithm. Logarithm

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 a degree by a rational exponent, to which the definition of a logarithm is reduced.
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 considering only numerical expressions, where it is not required to know the VA of the logarithm. All restrictions have already been taken into account by the authors of the problems. But when logarithmic equations and inequalities come into play, DL 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 look at the general scheme for 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 all! 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. It’s the same with decimal fractions: if you immediately convert them into ordinary 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;

Note also that the prime numbers themselves are always exact powers of themselves.

Decimal logarithm

Some logarithms are so common that they have a special name and 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. We are talking 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 is an irrational number; its exact value cannot be found and written 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.

This could be, for example, a calculator from the basic set of programs of the Windows operating system. The link to launch it is hidden quite in the main menu of the OS - open it by clicking on the “Start” button, then open its “Programs” section, go to the “Standard” subsection, and then to the “Utilities” section and, finally, click on the “Calculator” item " Instead of using the mouse and navigating through menus, you can use the keyboard and the program launch dialog - press the WIN + R key combination, type calc (this is the name of the calculator executable file) and press Enter.

Switch the calculator interface to advanced mode, which allows you to do... By default it opens in “normal” view, but you need “engineering” or “ ” (depending on the version of the OS you are using). Expand the “View” section in the menu and select the appropriate line.

Enter the argument whose natural value you want to evaluate. This can be done either from the keyboard or by clicking the corresponding buttons in the calculator interface on the screen.

Click the button labeled ln - the program will calculate the logarithm to base e and show the result.

Use one of the -calculators as an alternative to calculating the value of the natural logarithm. For example, the one located at http://calc.org.ua. Its interface is extremely simple - there is a single input field where you need to type the value of the number, the logarithm of which you need to calculate. Among the buttons, find and click the one that says ln. The script of this calculator does not require sending data to the server and a response, so you will receive the calculation result almost instantly. The only feature that should be taken into account is that the separator between the fractional and integer parts of the entered number must be a dot, and not .

The term " logarithm" comes from two Greek words, one meaning "number" and the other meaning "ratio". It denotes the mathematical operation of calculating a variable quantity (exponent) to which a constant value (base) must be raised to obtain the number indicated under the sign logarithm A. If the base is equal to a mathematical constant called the number "e", then logarithm called "natural".

You will need

• Internet access, Microsoft Office Excel or calculator.

Instructions

Use the many calculators available on the Internet - this is perhaps an easy way to calculate natural a. You don’t have to search for the appropriate service, since many search engines themselves have built-in calculators that are quite suitable for working with logarithm ami. For example, go to the main page of the largest online search engine - Google. No buttons are required here to enter values ​​or select functions; just enter the desired mathematical action in the query input field. Let's say, to calculate logarithm and the number 457 in base “e”, enter ln 457 - this will be enough for Google to display with an accuracy of eight decimal places (6.12468339) even without pressing the button to send a request to the server.

Use the appropriate built-in function if you need to calculate the value of a natural logarithm and occurs when working with data in the popular spreadsheet editor Microsoft Office Excel. This function is called here using the common notation logarithm and in uppercase - LN. Select the cell in which the calculation result should be displayed and enter an equal sign - this is how in this spreadsheet editor records should begin in the cells containing in the “Standard” subsection of the “All Programs” section of the main menu. Switch the calculator to a more functional mode by pressing Alt + 2. Then enter the value, natural logarithm which you want to calculate, and click in the program interface the button indicated by the symbols ln. The application will perform the calculation and display the result.

Video on the topic

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 (\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.)

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 - 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 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 a 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 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 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 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

Which 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's called natural because e is a universal measure of growth, so ln can be considered a universal way of 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

1.1. Determining the exponent for an integer exponent

X 1 = X
X 2 = X * X
X 3 = X * X * X
…
X N = X * X * … * X — N times

1.2. Zero degree.

By definition, it is generally accepted that the zero power of any number is 1:

1.3. Negative degree.

X -N = 1/X N

1.4. Fractional power, root.

X 1/N = N root of X.

For example: X 1/2 = √X.

1.5. Formula for adding powers.

X (N+M) = X N *X M

1.6.Formula for subtracting powers.

X (N-M) = X N /X M

1.7. Formula for multiplying powers.

X N*M = (X N) M

1.8. Formula for raising a fraction to a power.

(X/Y) N = X N /Y N

2. Number e.

The value of the number e is equal to the following limit:

E = lim(1+1/N), as N → ∞.

With an accuracy of 17 digits, the number e is 2.71828182845904512.

3. Euler's equality.

This equality connects five numbers that play a special role in mathematics: 0, 1, e, pi, imaginary unit.

E (i*pi) + 1 = 0

4. Exponential function exp(x)

exp(x) = e x

5. Derivative of exponential function

The exponential function has a remarkable property: the derivative of the function is equal to the exponential function itself:

(exp(x))" = exp(x)

6. Logarithm.

6.1. Definition of the logarithm function

If x = b y, then the logarithm is the function

Y = Log b(x).

The logarithm shows to what power a number must be raised - the base of the logarithm (b) to obtain a given number (X). The logarithm function is defined for X greater than zero.

For example: Log 10 (100) = 2.

6.2. Decimal logarithm

This is the logarithm to base 10:

Y = Log 10 (x) .

Denoted by Log(x): Log(x) = Log 10 (x).

An example of the use of the decimal logarithm is decibel.

6.3. Decibel

The item is highlighted on a separate page Decibel

6.4. Binary logarithm

This is the base 2 logarithm:

Y = Log 2 (x).

Denoted by Lg(x): Lg(x) = Log 2 (X)

6.5. Natural logarithm

This is the logarithm to base e:

Y = Log e (x) .

Denoted by Ln(x): Ln(x) = Log e (X)
The natural logarithm is the inverse function of the exponential function exp(X).

6.6. Characteristic points

Loga(1) = 0
Log a (a) = 1

6.7. Product logarithm formula

Log a (x*y) = Log a (x)+Log a (y)

6.8. Formula for logarithm of quotient

Log a (x/y) = Log a (x)-Log a (y)

6.9. Logarithm of power formula

Log a (x y) = y*Log a (x)

6.10. Formula for converting to a logarithm with a different base

Log b (x) = (Log a (x))/Log a (b)

Example:

Log 2 (8) = Log 10 (8)/Log 10 (2) =
0.903089986991943552 / 0.301029995663981184 = 3

7. Formulas useful in life

Often there are problems of converting volume into area or length and the inverse problem - converting area into volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be covered with boards contained in a certain volume, see calculation of boards, how many boards are in a cube. Or, if the dimensions of the wall are known, you need to calculate the number of bricks, see brick calculation.

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