So, we have powers of two. If you take the number from the bottom line, then you can easily find the power to which you have to raise a two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.
And now - in fact, the definition of the logarithm:
The base a logarithm of the argument x is the power to which the number a must be raised to get the number x .
Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.
For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, because 2 6 = 64.
The operation of finding the logarithm of a number to a given base is called the logarithm. So let's add a new row to our table:
| 2 1 | 2 2 | 2 3 | 2 4 | 2 5 | 2 6 |
| 2 | 4 | 8 | 16 | 32 | 64 |
| log 2 2 = 1 | log 2 4 = 2 | log 2 8 = 3 | log 2 16 = 4 | log 2 32 = 5 | log 2 64 = 6 |
Unfortunately, not all logarithms are considered so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.
Such numbers are called irrational: the numbers after the decimal point can be written indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100.
It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the basis is and where the argument is. To avoid annoying misunderstandings, just take a look at the picture:
Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.
We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:
- The argument and base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced.
- The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!
Such restrictions are called valid range(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.
Note that there are no restrictions on the number b (the value of the logarithm) is not imposed. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1 .
However, now we are considering only numerical expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions.
Now consider the general scheme for calculating logarithms. It consists of three steps:
- Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
- Solve the equation for the variable b: x = a b ;
- The resulting number b will be the answer.
That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.
Let's see how this scheme works with specific examples:
Task. Calculate the logarithm: log 5 25
- Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;
- Let's make and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2; - Received an answer: 2.
Task. Calculate the logarithm:
Task. Calculate the logarithm: log 4 64
- Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
- Let's make and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3; - Received an answer: 3.
Task. Calculate the logarithm: log 16 1
- Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
- Let's make and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0; - Received a response: 0.
Task. Calculate the logarithm: log 7 14
- Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;
- It follows from the previous paragraph that the logarithm is not considered;
- The answer is no change: log 7 14.
A small note to last example. How to make sure that a number is not an exact power of another number? Very simple - just expand it into prime factors. And if such factors cannot be collected in a degree with the same indicators, then the original number is not an exact degree.
Task. Find out if the exact powers of the number are: 8; 48; 81; 35; 14.
8 \u003d 2 2 2 \u003d 2 3 is the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 \u003d 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;
We also note that we prime numbers are always exact powers of themselves.
Decimal logarithm
Some logarithms are so common that they have a special name and designation.
The decimal logarithm of the x argument is the base 10 logarithm, i.e. the power to which you need to raise the number 10 to get the number x. Designation: lg x .
For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.
From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x
Everything that is true for ordinary logarithms is also true for decimals.
natural logarithm
There is another logarithm that has its own notation. In a sense, it is even more important than decimal. This is the natural logarithm.
The natural logarithm of the argument x is the logarithm to the 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 else is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers:
e = 2.718281828459...
We will not delve into what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x
Thus ln e = 1; log e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, unity: ln 1 = 0.
For natural logarithms, all the rules that are true for ordinary logarithms are valid.
\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
Let's explain it easier. For example, \(\log_(2)(8)\) is equal to the power \(2\) must be raised to to get \(8\). From this it is clear that \(\log_(2)(8)=3\).
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Examples: |
\(\log_(5)(25)=2\) |
because \(5^(2)=25\) |
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\(\log_(3)(81)=4\) |
because \(3^(4)=81\) |
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\(\log_(2)\)\(\frac(1)(32)\) \(=-5\) |
because \(2^(-5)=\)\(\frac(1)(32)\) |
Argument and base of the logarithm
Any logarithm has the following "anatomy":
The argument of the logarithm is usually written at its level, and the base is written in subscript closer to the sign of the logarithm. And this entry is read like this: "the logarithm of twenty-five to the base of five."
How to calculate the logarithm?
To calculate the logarithm, you need to answer the question: to what degree should the base be raised to get the argument?
For example, calculate the logarithm: a) \(\log_(4)(16)\) b) \(\log_(3)\)\(\frac(1)(3)\) c) \(\log_(\sqrt (5))(1)\) d) \(\log_(\sqrt(7))(\sqrt(7))\) e) \(\log_(3)(\sqrt(3))\)
a) To what power must \(4\) be raised to get \(16\)? Obviously the second. That's why:
\(\log_(4)(16)=2\)
\(\log_(3)\)\(\frac(1)(3)\) \(=-1\)
c) To what power must \(\sqrt(5)\) be raised to get \(1\)? And what degree makes any number a unit? Zero, of course!
\(\log_(\sqrt(5))(1)=0\)
d) To what power must \(\sqrt(7)\) be raised to get \(\sqrt(7)\)? In the first - any number in the first degree is equal to itself.
\(\log_(\sqrt(7))(\sqrt(7))=1\)
e) To what power must \(3\) be raised to get \(\sqrt(3)\)? From we know that is a fractional power, which means Square root is the degree \(\frac(1)(2)\) .
\(\log_(3)(\sqrt(3))=\)\(\frac(1)(2)\)
Example : Calculate the logarithm \(\log_(4\sqrt(2))(8)\)
Solution :
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\(\log_(4\sqrt(2))(8)=x\) |
We need to find the value of the logarithm, let's denote it as x. Now let's use the definition of the logarithm: |
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\((4\sqrt(2))^(x)=8\) |
What links \(4\sqrt(2)\) and \(8\)? Two, because both numbers can be represented by twos: |
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\(((2^(2)\cdot2^(\frac(1)(2))))^(x)=2^(3)\) |
On the left, we use the degree properties: \(a^(m)\cdot a^(n)=a^(m+n)\) and \((a^(m))^(n)=a^(m\cdot n)\) |
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\(2^(\frac(5)(2)x)=2^(3)\) |
The bases are equal, we proceed to the equality of indicators |
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\(\frac(5x)(2)\) \(=3\) |
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Multiply both sides of the equation by \(\frac(2)(5)\) |
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The resulting root is the value of the logarithm |
Answer : \(\log_(4\sqrt(2))(8)=1,2\)
Why was the logarithm invented?
To understand this, let's solve the equation: \(3^(x)=9\). Just match \(x\) to make the equality work. Of course, \(x=2\).
Now solve the equation: \(3^(x)=8\). What is x equal to? That's the point.
The most ingenious will say: "X is a little less than two." How exactly is this number to be written? To answer this question, they came up with the logarithm. Thanks to him, the answer here can be written as \(x=\log_(3)(8)\).
I want to emphasize that \(\log_(3)(8)\), as well as any logarithm is just a number. Yes, it looks unusual, but it is short. Because if we wanted to write it in the form decimal fraction, then it would look like this: \(1.892789260714.....\)
Example : Solve the equation \(4^(5x-4)=10\)
Solution :
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\(4^(5x-4)=10\) |
\(4^(5x-4)\) and \(10\) cannot be reduced to the same base. So here you can not do without the logarithm. Let's use the definition of the logarithm: |
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\(\log_(4)(10)=5x-4\) |
Flip the equation so x is on the left |
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\(5x-4=\log_(4)(10)\) |
Before us. Move \(4\) to the right. And don't be afraid of the logarithm, treat it like a regular number. |
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\(5x=\log_(4)(10)+4\) |
Divide the equation by 5 |
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\(x=\)\(\frac(\log_(4)(10)+4)(5)\) |
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Here is our root. Yes, it looks unusual, but the answer is not chosen. |
Answer : \(\frac(\log_(4)(10)+4)(5)\)
Decimal and natural logarithms
As stated in the definition of the logarithm, its base can be any positive number except one \((a>0, a\neq1)\). And among all the possible bases, there are two that occur so often that a special short notation was invented for logarithms with them:
Natural logarithm: a logarithm whose base is the Euler number \(e\) (equal to approximately \(2.7182818…\)), and the logarithm is written as \(\ln(a)\).
That is, \(\ln(a)\) is the same as \(\log_(e)(a)\)
Decimal logarithm: A logarithm whose base is 10 is written \(\lg(a)\).
That is, \(\lg(a)\) is the same as \(\log_(10)(a)\), where \(a\) is some number.
Basic logarithmic identity
Logarithms have many properties. One of them is called "Basic logarithmic identity" and looks like this:
| \(a^(\log_(a)(c))=c\) |
This property follows directly from the definition. Let's see how exactly this formula appeared.
Recall the short definition of the logarithm:
if \(a^(b)=c\), then \(\log_(a)(c)=b\)
That is, \(b\) is the same as \(\log_(a)(c)\). Then we can write \(\log_(a)(c)\) instead of \(b\) in the formula \(a^(b)=c\) . It turned out \(a^(\log_(a)(c))=c\) - the main logarithmic identity.
You can find the rest of the properties of logarithms. With their help, you can simplify and calculate the values of expressions with logarithms, which are difficult to calculate directly.
Example : Find the value of the expression \(36^(\log_(6)(5))\)
Solution :
Answer : \(25\)
How to write a number as a logarithm?
As mentioned above, any logarithm is just a number. The converse is also true: any number can be written as a logarithm. For example, we know that \(\log_(2)(4)\) is equal to two. Then you can write \(\log_(2)(4)\) instead of two.
But \(\log_(3)(9)\) is also equal to \(2\), so you can also write \(2=\log_(3)(9)\) . Similarly with \(\log_(5)(25)\), and with \(\log_(9)(81)\), etc. That is, it turns out
\(2=\log_(2)(4)=\log_(3)(9)=\log_(4)(16)=\log_(5)(25)=\log_(6)(36)=\ log_(7)(49)...\)
Thus, if we need, we can write the two as a logarithm with any base anywhere (even in an equation, even in an expression, even in an inequality) - we just write the squared base as an argument.
It's the same with a triple - it can be written as \(\log_(2)(8)\), or as \(\log_(3)(27)\), or as \(\log_(4)(64) \) ... Here we write the base in the cube as an argument:
\(3=\log_(2)(8)=\log_(3)(27)=\log_(4)(64)=\log_(5)(125)=\log_(6)(216)=\ log_(7)(343)...\)
And with four:
\(4=\log_(2)(16)=\log_(3)(81)=\log_(4)(256)=\log_(5)(625)=\log_(6)(1296)=\ log_(7)(2401)...\)
And with minus one:
\(-1=\) \(\log_(2)\)\(\frac(1)(2)\) \(=\) \(\log_(3)\)\(\frac(1)( 3)\) \(=\) \(\log_(4)\)\(\frac(1)(4)\) \(=\) \(\log_(5)\)\(\frac(1 )(5)\) \(=\) \(\log_(6)\)\(\frac(1)(6)\) \(=\) \(\log_(7)\)\(\frac (1)(7)\)\(...\)
And with one third:
\(\frac(1)(3)\) \(=\log_(2)(\sqrt(2))=\log_(3)(\sqrt(3))=\log_(4)(\sqrt( 4))=\log_(5)(\sqrt(5))=\log_(6)(\sqrt(6))=\log_(7)(\sqrt(7))...\)
Any number \(a\) can be represented as a logarithm with base \(b\): \(a=\log_(b)(b^(a))\)
Example : Find the value of an expression \(\frac(\log_(2)(14))(1+\log_(2)(7))\)
Solution :
Answer : \(1\)
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 not defined. Also, the base of the logarithm must be a positive number, 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 2.
Basic logarithmic identity
a log a b = b (a > 0, a ≠ 1) (2)It is important that the domains of definition of the right and left parts of this formula are 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" in solving equations and inequalities can lead to a change in the DPV.
Two obvious consequences of the definition of the 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.
The logarithm of the product and the 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 the thoughtless application of these formulas when solving logarithmic equations and inequalities. When they are used "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 restrict ourselves only to the case when f(x)>0 and g(x)>0. There is a narrowing of the range of admissible values, and this is categorically unacceptable, since it can lead to the 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 call for accuracy. 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! Taking the power out of the logarithm, we again narrow the ODZ. The reverse procedure leads to an expansion of the range of admissible values. All these remarks apply not only to the power of 2, but also to any even power.
Formula for moving to a new base
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 the conversion. If you have chosen the base c wisely (positive and not equal to 1), the formula for moving to a new base is perfectly safe.
If we choose the number b as a 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: lg2 + lg50.
Solution. lg2 + lg50 = lg100 = 2. We used the formula for the sum of logarithms (5) and the definition of the decimal logarithm.
Example 2 Calculate: lg125/lg5.
Solution. lg125/lg5 = log 5 125 = 3. We used the new base transition formula (8).
Table of formulas related to logarithms
| a log a b = b (a > 0, a ≠ 1) |
| log a a = 1 (a > 0, a ≠ 1) |
| log a 1 = 0 (a > 0, a ≠ 1) |
| log a (b c) = log a b + log a c (a > 0, a ≠ 1, b > 0, c > 0) |
| log a b c = log a b − log a c (a > 0, a ≠ 1, b > 0, c > 0) |
| log a b p = p log a b (a > 0, a ≠ 1, b > 0) |
| log a b = log c b log c a (a > 0, a ≠ 1, b > 0, c > 0, c ≠ 1) |
| log a b = 1 log b a (a > 0, a ≠ 1, b > 0, b ≠ 1) |
logarithms - traditional headache for many high school students. Especially - equations and inequalities with logarithms. For some reason, high school students do not like logarithms. And so they are afraid. And completely in vain.) For the logarithm itself is a very, very simple concept. Don't believe? See for yourself! In today's lesson.
So, let's go get acquainted.)
First, let's solve this very simple equation in our mind:
2 x = 4
This is the simplest exponential equation. It is so called due to the fact that the unknown X is in exponent. Even if you don't know how exponential equations, just in your mind choose x so that equality is fulfilled. Come on?! Surely, x = 2. Two squared is four.)
And now I will change only one number in it. Let's solve this equation now:
2 x = 5
And again we try to pick up X ...
What doesn't get picked? Two squared is four. Two cubed is eight. And we have five. They slipped past ... What to do? Just don't tell me that there is no such X! I will not believe.)
Agree that this is somehow unfair: with the four, the equation is solved in the mind, and with the five, it is no longer solved in any way. Mathematics does not accept such discrimination! For her, all numbers are equal partners.)
At this stage, we can only roughly estimate that x - some fractional number between two ( 2 2 = 4 ) and triple ( 2 3 = 8 ). We can even tinker a little with the calculator and approximately pick up, find this number. But such a fuss every time ... I agree, somehow sad ...
Math decides this problem very simple and elegant - introduction logarithm concepts.
So what is a logarithm? Let's go back to our mysterious equation:
2 x = 5
We comprehend the problem: we need to find a certain number X, to which you need to raise 2 to get 5 . Is this phrase clear? If not, read it again. And more... Until you realize. Because it is very important!
Here we call it mysterious number X logarithm of five to base two! In mathematical form, these words look like this:
X = log 2 5
And it's pronounced like this: "X is the logarithm of five to base two."
The number below (two) is called the base of the logarithm. It is written from below in the same way as in the exponential expression 2 x. It's very easy to remember.)
Well, that's all! We solved a terrible-looking exponential equation!
2 x = 5
X = log 2 5
And that's it! This is the correct and completely complete answer!
Maybe it bothers you that instead of a specific number, I write some strange letters and icons?
Well, okay, we persuaded ... Especially for you:
X = log 2 5 = 2.321928095…
Keep in mind that this number never ends. Yes Yes! It's irrational...
Here is the answer to your question, what are logarithms for?. We need logarithms, first of all, to solve exponential equations! Those that are not solved at all without logarithms ...
For example, solving the exponential equation
3x=9
You can forget about logarithms. It is immediately clear that x = 2.
But, solving the equation, let's say this
3 x = 7,
You approximately get this shaggy response:
X ≈ 1.77124375
But through the logarithm is given pitch perfect answer:
X = log 3 7.
And that's all.) That's why they write logarithms instead of ugly irrational numbers. Who needs a numerical answer - he will count on a calculator or at least in Excel.) And earlier, when there were no calculators and computers, there were special tables of logarithms. Bulky and hefty. Just like the Bradys tables for sines and cosines. And even this tool was - logarithmic ruler. Which allowed with good accuracy to calculate a lot of useful things. And not just logarithms.)
Here you go. Now, imperceptibly for ourselves, we have learned to decide All exponential equations of this brutal type.
For example:
2 x = 13
No problem:
X = log 2 13
5 x = 26
Also elementary!
X = log 5 26
11 x = 0.123
And this is not a question:
X = log 11 0.123
These are all correct answers! Well, how? Tempting, right?
Now let's think about the meaning of the operation of finding the logarithm.
As we know, for every action mathematicians try to find a reaction (i.e. reverse action). For addition it is subtraction, for multiplication it is division. What is the reverse action for exponentiation?
Let's get a look. What are our main operating figures when raising to a power? Here they are:
a n = b
a - base,
n - index,
b - the degree itself.
Now let's think: if we know degree(b) and known index this very degree (n), but you need to find base (a) , then what do we usually do? Right! We extract the root nth degree! Like this:
Now let's look at another situation: we again know degree(b), but this time instead of the exponent n we know base(a), but you just need to find this very indicator (n). What we are going to do?
This is where logarithms come to the rescue! They write exactly like this:
"En" (n) is the number to be raised to "a", To obtain "b". That's all. That's the whole point of the logarithm. The operation of finding the logarithm is just a search indicator degrees in famous degrees And foundation.
Thus, for exponentiation in mathematics, there is two different nature reverse action. This root extraction And finding the logarithm. But, let's say for multiplication, there is only one inverse action - division. It is understandable: any of the unknown factors - which is the first, which is the second - is sought using a single operation - division.)
The simplest examples with logarithms.
Now the news is not good. If the logarithm is considered exactly, then its must be considered, Yes.
Let's say if somewhere in the equation you got
x = log 3 9 ,
That answer will not be appreciated. We need to calculate the logarithm and write down:
x = 2
And how did we understand that log 3 9=2? We translate equality from mathematical language into Russian: the logarithm of nine to the base of three is the number to which three must be raised to get nine. And what number do you need to raise a three to get a nine? Well, of course! Must be squared. That is, two.)
And what is, say, log 5 125? And to what extent does five give us 125? In the third, of course (i.e. in a cube)!
So log 5 125 = 3.
Log 7 7 = ?
To what power must 7 be raised to get 7? First!
Here is your answer: log 7 7 = 1
How about an example like this?
Log 3 1 = ?
And to what power must three be raised to get one? Didn't you guess? Do you remember .) Yes! To zero! Here we write:
Log 3 1 = 0
Got the principle? Then we train:
Log 2 16 = …
Log 4 64 = …
Log 13 13 = …
Log 3 243 = …
Log 15 1 = …
Answers (in disarray): 1; 3; 5; 0; 4.
What? Forgot to what extent 3 gives 243? Well, there's nothing to be done: the degrees of popular numbers must be recognized. In face! Well, the multiplication table is a reliable companion and assistant. And not only in logarithms.)
Well, quite simple examples have been solved, and now we are stepping up a notch. We recall negative and fractional indicators.)
Let's solve this example:
Log 4 0.25 = ?
Hmm ... And to what power do you need to raise the four to get 0.25? So you can't tell right off the bat. If you work only with natural indicators. But degrees in mathematics, as you know, are not only natural. It's time to connect our knowledge about negative indicators and remember that
0,25 = 1/4 = 4 -1
Therefore, we can safely write:
Log 4 0.25 = log 4 4 -1 = -1.
And that's it.)
Another example:
Log 4 2 = ?
To what power do you need to raise 4 to get 2? To answer this question, we will have to connect our knowledge of the roots. And remember that the deuce is square root of four:
And the root square math lets represent as a degree! With an indicator of 1/2. So we write:
So our logarithm will be:
Well, congratulations! Here we are with you and met with logarithms. At the most primitive initial level.) And you yourself saw for yourself that they are not at all as scary as you might have thought before. But logarithms, like any other mathematical concepts, have their own properties and their own special features. About both (about properties and about chips) - in the next lesson.
And now we decide on our own.
Calculate:
Answers (in disarray): 4.4; 0; 1; 6; 4; 2.
LOGARITHM
a number that simplifies many complex arithmetic operations. Using their logarithms instead of numbers in calculations makes it possible to replace multiplication with a simpler operation of addition, division with subtraction, raising to a power with multiplication, and extracting roots with division. general description. The logarithm of a given number is the exponent to which another number, called the base of the logarithm, must be raised to get the given number. For example, the base 10 logarithm of 100 is 2. In other words, 10 must be squared to get 100 (102 = 100). If n is a given number, b is a base, and l is a logarithm, then bl = n. The number n is also called the antilogarithm to the base b of the number l. For example, the antilogarithm of 2 to base 10 is 100. This can be written as logb n = l and antilogb l = n. The main properties of logarithms:
Any positive number other than one can serve as the base of logarithms, but, unfortunately, it turns out that if b and n are rational numbers, then in rare cases there is a rational number l such that bl = n. However, it is possible to define an irrational number l, for example, such that 10l = 2; this irrational number l can be approximated with any required accuracy rational numbers. It turns out that in the example above, l is approximately equal to 0.3010, and this approximate value of the logarithm to the base 10 of the number 2 can be found in four-digit tables of decimal logarithms. Base 10 logarithms (or decimal logarithms) are used so often in calculations that they are called ordinary logarithms and are written as log2 = 0.3010 or log2 = 0.3010, omitting the explicit indication of the base of the logarithm. Logarithms to the base e, a transcendental number approximately equal to 2.71828, are called natural logarithms. They are found predominantly in 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 e0.6931 = 2.
see also NUMBER e . Using tables of ordinary logarithms. The ordinary logarithm of a number is the exponent to which you need to raise 10 to get the given number. Since 100 = 1, 101 = 10, and 102 = 100, we immediately get that log1 = 0, log10 = 1, log100 = 2, and so on. for increasing integer powers of 10. Similarly, 10-1 = 0.1, 10-2 = 0.01 and hence log0.1 = -1, log0.01 = -2, and so on. for all negative integer powers of 10. The usual logarithms of the remaining numbers are enclosed between the logarithms of the nearest integer powers of 10; log2 must be enclosed between 0 and 1, log20 between 1 and 2, and log0.2 between -1 and 0. Thus, the logarithm has two parts, an integer and a decimal enclosed between 0 and 1. The integer part is called the characteristic of the logarithm and is determined by the number itself, the fractional part is called the mantissa and can be found from the tables. Also, log20 = log(2´10) = log2 + log10 = (log2) + 1. The logarithm of 2 is 0.3010, so log20 = 0.3010 + 1 = 1.3010. Similarly, log0.2 = log(2e10) = log2 - log10 = (log2) - 1 = 0.3010 - 1. By subtracting, we get log0.2 = - 0.6990. However, it is more convenient to represent log0.2 as 0.3010 - 1 or as 9.3010 - 10; can be formulated and general rule: all numbers obtained from a given number by multiplying by a power of 10 have the same mantissa equal to the mantissa given number. In most tables, the mantissas of numbers ranging from 1 to 10 are given, since the mantissas of all other numbers can be obtained from those given in the table. Most tables give logarithms to four or five decimal places, although there are seven-digit tables and tables with more a large number signs. Learning how to use such tables is easiest with examples. To find log3.59, first of all, note that the number 3.59 is between 100 and 101, so its characteristic is 0. We find the number 35 in the table (on the left) and move along the row to the column that has the number 9 on top; the intersection of this column and row 35 is 5551, so log3.59 = 0.5551. To find the mantissa of a number with four significant digits, you need to resort to interpolation. In some tables, interpolation is facilitated by the proportional parts given in the last nine columns on the right side of each table page. Find now log736.4; the number 736.4 lies between 102 and 103, so the characteristic of its logarithm is 2. In the table we find the row to the left of which is 73 and column 6. At the intersection of this row and this column is the number 8669. Among the linear parts we find column 4. On the intersection of row 73 and column 4 is the number 2. Adding 2 to 8669, we get the mantissa - it is equal to 8671. Thus, log736.4 = 2.8671.
natural logarithms. Tables and properties of natural logarithms are similar to tables and properties of ordinary logarithms. The main difference between the two is that the integer part of the natural logarithm is not significant in determining the position of the decimal point, and therefore the difference between the mantissa and the characteristic does not play a special role. Natural logarithms of numbers 5.432; 54.32 and 543.2 are, respectively, 1.6923; 3.9949 and 6.2975. The relationship between these logarithms becomes apparent if we consider the differences between them: log543.2 - log54.32 = 6.2975 - 3.9949 = 2.3026; the last number is nothing but the natural logarithm of the number 10 (written like this: ln10); log543.2 - log5.432 = 4.6052; the last number is 2ln10. But 543.2 = 10*54.32 = 102*5.432. Thus, by the natural logarithm of a given number a, one can find the natural logarithms of numbers equal to the products of the number a and any powers of n of the number 10, if ln10 multiplied by n is added to lna, i.e. ln(a*10n) = lna + nln10 = lna + 2.3026n. For example, ln0.005432 = ln(5.432*10-3) = ln5.432 - 3ln10 = 1.6923 - (3*2.3026) = - 5.2155. Therefore, tables of natural logarithms, like tables of ordinary logarithms, usually contain only the logarithms of numbers from 1 to 10. In the system of natural logarithms, one can speak of antilogarithms, but more often one speaks of an exponential function or an exponential. If x = lny, then y = ex, and y is called the exponent of x (for typographical convenience, y = exp x is often written). 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 logb a = x, then bx = a, and hence logc bx = logc a or xlogc b = logc a, or x = logc a/logc b = logb a. Therefore, using this inversion formula from a table of logarithms to base c, one can construct tables of logarithms to any other base b. The factor 1/logc b is called the modulus of the transition from base c to base b. Nothing prevents, for example, using the inversion formula, or the transition from one system of logarithms to another, to find natural logarithms from the table of ordinary logarithms or to make the reverse transition. For example, log105.432 = loge 5.432/loge 10 = 1.6923/2.3026 = 1.6923 x 0.4343 = 0.7350. The number 0.4343, by which the natural logarithm of a given number must be multiplied to obtain the ordinary logarithm, is the modulus of the transition to the system of ordinary logarithms.
Special tables. Logarithms were originally invented to use their properties logab = loga + logb and loga/b = loga - logb to convert products into sums and quotients into differences. In other words, if loga and logb are known, then with the help of addition and subtraction we can easily find the logarithm of the product and the quotient. In astronomy, however, often set values loga and logb need to find log(a + b) or log(a - b). Of course, it would be possible to first find a and b from the tables of logarithms, then perform the indicated addition or subtraction and, again referring to the tables, find the required logarithms, but such a procedure would require three visits to the tables. Z. Leonelli in 1802 published tables of the so-called. Gaussian logarithms - the logarithms of addition of sums and differences - which made it possible to confine ourselves to one recourse to tables. In 1624, I. Kepler proposed tables of proportional logarithms, i.e. logarithms of numbers a/x, where a is some positive constant. These tables are used primarily by astronomers and navigators. Proportional logarithms for a = 1 are called logarithms and are used in calculations when you have to deal with products and quotients. The logarithm of the number n is equal to the logarithm of the reciprocal of the number; those. cologn = log1/n = - logn. If log2 = 0.3010, then colog2 = - 0.3010 = 0.6990 - 1. The advantage of using logarithms is that when calculating the value of the logarithm of expressions like pq/r, the triple sum of the positive decimals of logp + logq + cologr is easier to find than the mixed sum and difference logp + logq - logr.
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 integers positive degrees whole numbers were used to calculate compound interest. Much later, Archimedes (287-212 BC) used the powers of 108 to find an upper limit on the number of grains of sand needed to completely fill the universe known at that time. Archimedes drew attention to the property of the exponents that underlies the effectiveness of logarithms: the product of the powers corresponds to the sum of the exponents. At the end of the Middle Ages and the beginning of the New Age, mathematicians increasingly began to refer to the relationship between geometric and arithmetic progressions. M. Stiefel in his essay Arithmetic of Integers (1544) gave a table of positive and negative powers of the number 2:
Stiefel noticed that the sum of the two numbers in the first row (the row of exponents) is equal to the exponent of two, which corresponds to the product of the two corresponding numbers in the bottom row (the row of exponents). In connection with this table, Stiefel formulated four rules that are equivalent to the four modern rules for operations on exponents or four rules for operations on logarithms: the sum in the top row corresponds to the product in the bottom row; the subtraction in the top row corresponds to the division in the bottom row; multiplication in the top row corresponds to exponentiation in the bottom row; the division in the top row corresponds to the root extraction in the bottom row. Apparently, rules similar to those of Stiefel led J. Napier to formally introduce the first system of logarithms in the Description of the amazing table of logarithms, published in 1614. But Napier's thoughts have been occupied with the problem of converting products into sums since more than Ten years before the publication of his work, Napier received news from Denmark that at Tycho Brahe's observatory his assistants had a method for converting products into sums. The method mentioned in Napier's communication was based on the use of trigonometric formulas of the type
Therefore Napier's tables consisted mainly of the logarithms of trigonometric functions. Although the concept of base was not explicitly included in the definition proposed by Napier, the number equivalent to the base of the system of logarithms in his system was played by the number (1 - 10-7)ґ107, approximately equal to 1/e. Independently of Napier and almost simultaneously with him, a system of logarithms, quite similar in type, was invented and published by J. Burgi in Prague, who published the Tables of Arithmetic and Geometric Progressions in 1620. These were tables of antilogarithms in base (1 + 10-4)*10 4, a fairly good approximation of the number e. In Napier's system, the logarithm of the number 107 was taken as zero, and as the numbers decreased, the logarithms increased. When G. Briggs (1561-1631) visited Napier, both agreed that it would be more convenient to use the number 10 as the base and take the logarithm of unity zero. Then, as the numbers increase, their logarithms would increase. Thus 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 Napier, are often called non-Pier. The terms "characteristic" and "mantissa" were proposed by Briggs. First logarithms in effect historical reasons used approximations to the numbers 1/e and e. Somewhat later, the idea of natural logarithms was associated with the study of areas under the hyperbola xy = 1 (Fig. 1). In the 17th century it was shown that the area bounded by this curve, the x axis and the ordinates x = 1 and x = a (in Fig. 1 this area is covered with thicker and rarer dots) increases in arithmetic progression, when a increases in geometric progression. It is this dependence that arises in the rules for actions on exponents and logarithms. This gave grounds to call the Napier logarithms "hyperbolic logarithms".
Logarithmic function. There was a time when logarithms were considered solely as a means of calculation, but in the 18th century, mainly due to the work of Euler, the concept was formed logarithmic function. The graph of such a function y = lnx, whose ordinates increase in arithmetic progression, while the abscissas increase in geometric progression, is shown in Fig. 2a. The graph of the inverse, or exponential (exponential) function y = ex, whose ordinates increase exponentially, and the abscissas - arithmetic, is presented, respectively, in Fig. 2b. (The curves y = logx and y = 10x are similar in shape to the curves y = lnx and y = ex.) Alternative definitions of the logarithmic function have also been proposed, for example,
Thanks to the work of Euler, the relations between logarithms and trigonometric functions in the complex plane. From the identity eix = cos x + i sin x (where the angle x is measured in radians), Euler concluded that every non-zero real number has infinitely many natural logarithms; they are all complex in the case negative numbers and all but one, in the case positive numbers. Since eix = 1 not only for x = 0, but also for x = ± 2kp, where k is any positive integer, any of the numbers 0 ± 2kpi can be taken as the natural logarithm of the number 1; and, similarly, the natural logarithms of -1 are complex numbers of the form (2k + 1)pi, where k is an integer. Similar statements are also true for general logarithms or other systems of logarithms. In addition, the definition of logarithms can be generalized using the Euler identities to include the complex logarithms of complex numbers. An alternative definition of the logarithmic function is provided by functional analysis. If f(x) - continuous function real number x having the following three properties: f(1) = 0, f(b) = 1, f(uv) = f(u) + f(v), then f(x) is defined as the logarithm of x to the base b. This definition has a number of advantages over the definition given at the beginning of this article.
Applications. Logarithms were originally used solely to simplify calculations, and this application is still one of their most important. The calculation of products, quotients, powers and roots is facilitated not only by the wide availability of published tables of logarithms, but also by the use of the so-called. slide rule - a computing tool, the principle of which is based on the properties of logarithms. The ruler is equipped with logarithmic scales, i.e. the distance from the number 1 to any number x is chosen to be log x; by shifting one scale relative to another, it is possible to plot the sums or differences of logarithms, which makes it possible to read products or partials of the corresponding numbers directly from the scale. To take advantage of the presentation of numbers in a logarithmic form allows the so-called. logarithmic paper for plotting (paper with logarithmic scales printed on it along both coordinate axes). If 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 is an equation linear in log y and log x. On the contrary, if the logarithmic graph of some functional dependence has the form of a straight line, then this dependence is a power law. Semi-logarithmic paper (where the y-axis is on a logarithmic scale and the abscissa is on a uniform scale) is useful when exponential functions need to be identified. Equations of the form y = kbrx arise whenever a quantity, such as the population, the amount of radioactive material, or the bank balance, decreases or increases at a rate proportional to the available this moment the number of inhabitants, radioactive material or money. If such a dependence is applied to semi-logarithmic paper, then the graph will look like a straight line. The logarithmic function arises in connection with a variety of natural forms. Flowers in sunflower inflorescences line up in logarithmic spirals, the shells of the Nautilus mollusk, the horns of the mountain sheep and the beaks of parrots are twisted. All these natural forms are examples of the curve known as the logarithmic spiral because its equation in polar coordinates is r = aebq, or lnr = lna + bq. Such a curve is described by a moving point, the distance from the pole of which grows exponentially, and the angle described by its radius vector grows arithmetic. The ubiquity of such a curve, and consequently of the logarithmic function, is well illustrated by the fact that it occurs in regions as far away and quite different as the contour of the eccentric cam and the trajectory of certain insects flying towards the light.
Collier Encyclopedia. - Open society. 2000 .
See what "LOGARIFM" is in other dictionaries:
- (Greek, from logos relation, and arithmos number). The number of an arithmetic progression corresponding to the number of a geometric progression. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. LOGARIFM Greek, from logos, relation, ... ... Dictionary of foreign words of the Russian language
The given number N at the base a is the exponent of the power of y to which you need to raise the number a to get N; thus, N = ay. The logarithm is usually denoted by logaN. Logarithm with base e? 2.718... is called natural and denoted by lnN.… … Big Encyclopedic Dictionary
- (from the Greek logos ratio and arithmos number) numbers N in base a (O ... Modern Encyclopedia
