Logarithmic Expressions, solving examples. In this article we will look at problems related to solving logarithms. The tasks ask the question of finding the meaning of an expression. It should be noted that the concept of logarithm is used in many tasks and understanding its meaning is extremely important. As for the Unified State Examination, the logarithm is used when solving equations, in applied problems, also in tasks related to the study of functions.
Let us give examples to understand the very meaning of the logarithm:
Basic logarithmic identity:
Properties of logarithms that must always be remembered:
*Logarithm of the product equal to the sum logarithms of factors.
* * *
*The logarithm of a quotient (fraction) is equal to the difference between the logarithms of the factors.
* * *
*Logarithm of degree equal to the product exponent by the logarithm of its base.
* * *
*Transition to a new foundation
* * *
More properties:
* * *
The calculation of logarithms is closely related to the use of properties of exponents.
Let's list some of them:
The essence of this property lies in the fact that when transferring the numerator to the denominator and vice versa, the sign of the exponent changes to the opposite. For example:
A corollary from this property:
* * *
When raising a power to a power, the base remains the same, but the exponents are multiplied.
* * *
As you have seen, the concept of a logarithm itself is simple. The main thing is what is needed good practice, which gives a certain skill. Of course, knowledge of formulas is required. If the skill in converting elementary logarithms has not been developed, then when solving simple tasks you can easily make a mistake.
Practice, solve the simplest examples from the mathematics course first, then move on to more complex ones. In the future, I will definitely show how “ugly” logarithms are solved; these won’t appear on the Unified State Examination, but they are of interest, don’t miss them!
That's all! Good luck to you!
Sincerely, Alexander Krutitskikh
P.S: I would be grateful if you tell me about the site on social networks.
Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, address Email etc.
How we use your personal information:
- Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes such as auditing, data analysis and various studies in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary, in accordance with the law, judicial procedure, V trial, and/or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
Let's explain it more simply. For example, \(\log_(2)(8)\) equal to the power, to which \(2\) must be raised to obtain \(8\). From this it is clear that \(\log_(2)(8)=3\).
|
Examples: |
\(\log_(5)(25)=2\) |
because \(5^(2)=25\) |
||
|
\(\log_(3)(81)=4\) |
because \(3^(4)=81\) |
|||
|
\(\log_(2)\)\(\frac(1)(32)\) \(=-5\) |
because \(2^(-5)=\)\(\frac(1)(32)\) |
Argument and base of logarithm
Any logarithm has the following “anatomy”:
The argument of a logarithm is usually written at its level, and the base is written in subscript closer to the logarithm sign. And this entry reads like this: “logarithm of twenty-five to base five.”
How to calculate logarithm?
To calculate the logarithm, you need to answer the question: to what power 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 one. That's why:
\(\log_(4)(16)=2\)
\(\log_(3)\)\(\frac(1)(3)\) \(=-1\)
c) To what power must \(\sqrt(5)\) be raised to get \(1\)? What power makes any number one? Zero, of course!
\(\log_(\sqrt(5))(1)=0\)
d) To what power must \(\sqrt(7)\) be raised to obtain \(\sqrt(7)\)? Firstly, any number to the first power is equal to itself.
\(\log_(\sqrt(7))(\sqrt(7))=1\)
e) To what power must \(3\) be raised to obtain \(\sqrt(3)\)? From we know what it is fractional power, and that means Square root is the power of \(\frac(1)(2)\) .
\(\log_(3)(\sqrt(3))=\)\(\frac(1)(2)\)
Example : Calculate logarithm \(\log_(4\sqrt(2))(8)\)
Solution :
|
\(\log_(4\sqrt(2))(8)=x\) |
We need to find the value of the logarithm, let's denote it as x. Now let's use the definition of a logarithm: |
|
|
\((4\sqrt(2))^(x)=8\) |
What connects \(4\sqrt(2)\) and \(8\)? Two, because both numbers can be represented by twos: |
|
|
\(((2^(2)\cdot2^(\frac(1)(2))))^(x)=2^(3)\) |
On the left we use the properties of the degree: \(a^(m)\cdot a^(n)=a^(m+n)\) and \((a^(m))^(n)=a^(m\cdot n)\) |
|
|
\(2^(\frac(5)(2)x)=2^(3)\) |
The bases are equal, we move on to equality of indicators |
|
|
\(\frac(5x)(2)\) \(=3\) |
|
Multiply both sides of the equation by \(\frac(2)(5)\) |
|
|
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 equation work. Of course, \(x=2\).
Now solve the equation: \(3^(x)=8\).Why equal to x? That's the point.
The smartest ones will say: “X is a little less than two.” How exactly to write this number? To answer this question, the logarithm was invented. Thanks to him, the answer here can be written as \(x=\log_(3)(8)\).
I want to emphasize that \(\log_(3)(8)\), like any logarithm is just a number. Yes, it looks unusual, but it’s short. Because if we wanted to write it in the form decimal, then it would look like this: \(1.892789260714.....\)
Example : Solve the equation \(4^(5x-4)=10\)
Solution :
|
\(4^(5x-4)=10\) |
\(4^(5x-4)\) and \(10\) cannot be brought to the same base. This means you can’t do without a logarithm. Let's use the definition of logarithm: |
|
|
\(\log_(4)(10)=5x-4\) |
Let's flip the equation so that X is on the left |
|
|
\(5x-4=\log_(4)(10)\) |
Before us. Let's move \(4\) to the right. And don't be afraid of the logarithm, treat it like an ordinary number. |
|
|
\(5x=\log_(4)(10)+4\) |
Divide the equation by 5 |
|
|
\(x=\)\(\frac(\log_(4)(10)+4)(5)\) |
|
This is our root. Yes, it looks unusual, but they don’t choose the answer. |
Answer : \(\frac(\log_(4)(10)+4)(5)\)
Decimal and natural logarithms
As stated in the definition of a logarithm, its base can be any positive number, except for the unit \((a>0, a\neq1)\). 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 Euler's 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 the “Basic Logarithmic Identity” and looks like this:
| \(a^(\log_(a)(c))=c\) |
This property follows directly from the definition. Let's see exactly how this formula came about.
Let us recall a short notation of the definition of 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 other 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 instead of two you can write \(\log_(2)(4)\).
But \(\log_(3)(9)\) is also equal to \(2\), which means we can also write \(2=\log_(3)(9)\) . Likewise 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 two as a logarithm with any base anywhere (be it in an equation, in an expression, or in an inequality) - we simply write the base squared as an argument.
It’s the same with the 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 meaning of the expression \(\frac(\log_(2)(14))(1+\log_(2)(7))\)
Solution :
Answer : \(1\)
Logarithm of a number N based on A called exponent X , to which you need to build A to get the number N
Provided that 
,
,
From the definition of logarithm it follows that 
, i.e.
- this equality is fundamental logarithmic identity.
Logarithms to base 10 are called decimal logarithms. Instead of 
write 
.
Logarithms to the base e
are called natural and are designated 
.
Basic properties of logarithms.
The logarithm of one is equal to zero for any base.
The logarithm of the product is equal to the sum of the logarithms of the factors.
3) The logarithm of the quotient is equal to the difference of the logarithms
Factor 
called the modulus of transition from logarithms to the base a
to logarithms at the base b
.
Using properties 2-5, it is often possible to reduce the logarithm of a complex expression to the result of simple arithmetic operations on logarithms.
For example, 
Such transformations of a logarithm are called logarithms. Transformations inverse to logarithms are called potentiation.
Chapter 2. Elements of higher mathematics.
1. Limits
Limit of the function 
is a finite number A if, as xx
0
for each predetermined 
, there is such a number 
that as soon as 
, That 
.
A function that has a limit differs from it by an infinitesimal amount: 
, where- b.m.v., i.e. 
.
Example. Consider the function 
.
When striving 
, function y
tends to zero: 
1.1. Basic theorems about limits.
Limit constant value equal to this constant value
.
Amount (difference) limit finite number functions is equal to the sum (difference) of the limits of these functions.
The limit of the product of a finite number of functions is equal to the product of the limits of these functions.
The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is not zero.
Wonderful Limits
,
, Where 
1.2. Limit Calculation Examples
However, not all limits are calculated so easily. More often, calculating the limit comes down to revealing an uncertainty of the type: 
or .
.
2. Derivative of a function
Let us have a function 
, continuous on the segment 
.
Argument 
got some increase 
. Then the function will receive an increment 
.
Argument value 
corresponds to the function value 
.
Argument value 
corresponds to the function value.
Hence, .
Let us find the limit of this ratio at 
. If this limit exists, then it is called the derivative of the given function.
Definition 3 Derivative of a given function 
by argument 
is called the limit of the ratio of the increment of a function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.
Derivative of a function 
can be designated as follows:
;
;
;
.
Definition 4The operation of finding the derivative of a function is called differentiation.
2.1. Mechanical meaning of derivative.
Let us consider the rectilinear motion of some rigid body or material point.
Let at some point in time 
moving point 
was at a distance 
from the starting position 
.
After some period of time 
she moved a distance 
. Attitude 
=
- average speed material point 
. Let us find the limit of this ratio, taking into account that 
.
Therefore, the definition instantaneous speed motion of a material point comes down to finding the derivative of the path with respect to time.
2.2. Geometric meaning derivative
Let us have a graphically defined function 
.
Rice. 1. Geometric meaning of derivative
If 
, then point 
, will move along the curve, approaching the point 
.
Hence 
, i.e. the value of the derivative for a given value of the argument 
numerically equal to the tangent of the angle formed by the tangent at a given point with the positive direction of the axis 
.
2.3. Table of basic differentiation formulas.
Power function
|
|
|
|
|
|
|
Exponential function
|
|
|
|
|
Logarithmic function
|
|
|
|
|
Trigonometric function
|
|
|
|
|
|
|
|
|
|
Inverse trigonometric function
|
|
|
|
|
|
|
|
|
|
2.4. Rules of differentiation.
Derivative of 
Derivative of the sum (difference) of functions
Derivative of the product of two functions
Derivative of the quotient of two functions
2.5. Derivative of complex function.
Let the function be given 
such that it can be represented in the form
And 
, where the variable 
is an intermediate argument, then 
The derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument with respect to x.
Example 1. 
Example 2. 
3. Differential function.
Let there be 
, differentiable on some interval 
let it go at
this function has a derivative
,
then we can write
(1),
Where 
- an infinitesimal quantity,
since when 
Multiplying all terms of equality (1) by 
we have:
Where 
- b.m.v. higher order.
Magnitude 
called the differential of the function 
and is designated 
.
3.1. Geometric value of the differential.
Let the function be given 
.
Fig.2. Geometric meaning of differential.
.
Obviously, the differential of the function 
is equal to the increment of the ordinate of the tangent at a given point.
3.2. Derivatives and differentials of various orders.
If there 
, Then 
is called the first derivative.
The derivative of the first derivative is called the second-order derivative and is written 
.
Derivative of the nth order of the function 
is called the (n-1)th order derivative and is written:
.
The differential of the differential of a function is called the second differential or second order differential.
.
.
3.3 Solving biological problems using differentiation.
Task 1. Studies have shown that the growth of a colony of microorganisms obeys the law 
, Where N
– number of microorganisms (in thousands), t
– time (days).
b) Will the population of the colony increase or decrease during this period?
Answer. The size of the colony will increase.
Task 2. The water in the lake is periodically tested to monitor the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio
.
When will the lake have a minimum concentration of bacteria and will it be possible to swim in it?
Solution: A function reaches max or min when its derivative is zero.
,
Let's determine the max or min will be in 6 days. To do this, let's take the second derivative.
Answer: After 6 days there will be a minimum concentration of bacteria.
(from Greek λόγος - “word”, “relation” and ἀριθμός - “number”) numbers b based on a(log α b) is called such a number c, And b= a c, that is, records log α b=c And b=ac are equivalent. The logarithm makes sense if a > 0, a ≠ 1, b > 0.
In other words logarithm numbers b based on A formulated as an exponent to which a number must be raised a to get the number b(logarithm exists only for positive numbers).
From this formulation it follows that the calculation x= log α b, is equivalent to solving the equation a x =b.
For example:
log 2 8 = 3 because 8 = 2 3 .
Let us emphasize that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value, when the number under the logarithm sign acts as a certain power of the base. Indeed, the formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b based on a equals With. It is also clear that the topic of logarithms is closely related to the topic powers of a number.
Calculating the logarithm is called logarithm. Logarithm is mathematical operation taking the logarithm. When taking logarithms, products of factors are transformed into sums of terms.
Potentiation is the inverse mathematical operation of logarithm. During potentiation, a given base is raised to the degree of expression over which potentiation is performed. In this case, the sums of terms are transformed into a product of factors.
Real logarithms with bases 2 (binary) are quite often used, e Euler number e ≈ 2.718 ( natural logarithm) and 10 (decimal).
On at this stage it is advisable to consider logarithm samples log 7 2 , ln √ 5, lg0.0001.
And the entries lg(-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the logarithm sign, in the second - a negative number in the base, and in the third - both a negative number under the logarithm sign and a unit in the base.
Conditions for determining the logarithm.
It is worth considering separately the conditions a > 0, a ≠ 1, b > 0.under which we get definition of logarithm. Let's consider why these restrictions were taken. An equality of the form x = log α will help us with this b, called the basic logarithmic identity, which directly follows from the definition of logarithm given above.
Let's take the condition a≠1. Since one to any power is equal to one, then the equality x=log α b can only exist when b=1, but log 1 1 will be any real number. To eliminate this ambiguity, we take a≠1.
Let us prove the necessity of the condition a>0. At a=0 according to the formulation of the logarithm can exist only when b=0. And accordingly then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be eliminated by the condition a≠0. And when a<0 we would have to reject the analysis of rational and irrational values of the logarithm, since a degree with a rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition is stipulated a>0.
AND last condition b>0 follows from inequality a>0, since x=log α b, and the value of the degree with a positive base a always positive.
Features of logarithms.
Logarithms characterized by distinctive features, which led to their widespread use to significantly facilitate painstaking calculations. When moving “to the world of logarithms,” multiplication is transformed by much more easy folding, division is subtraction, and exponentiation and root extraction are transformed, respectively, into multiplication and division by the exponent.
Formulation of logarithms and table of their values (for trigonometric functions) was first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, enlarged and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until the use of electronic calculators and computers.