Reducing logarithms with the same base. Logarithm rules for operating with logarithms
The basic properties of the natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, power series expansion and representation of the function ln x using complex numbers are given.
Definition
Natural logarithm is the function y = ln x, the inverse of the exponential, x = e y, and is the logarithm to the base of the number e: ln x = log e x.
The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.
Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.
Graph of the function y = ln x.
Graph of natural logarithm (functions y = ln x) is obtained from the exponential graph by mirror reflection relative to the straight line y = x.
The natural logarithm is defined for positive values of the variable x.
It increases monotonically in its domain of definition. 0 At x →
the limit of the natural logarithm is minus infinity (-∞).
As x → + ∞, the limit of the natural logarithm is plus infinity (+ ∞). For large x, the logarithm increases quite slowly. Any power function x a with a positive exponent a grows faster than the logarithm.
Properties of the natural logarithm
Domain of definition, set of values, extrema, increase, decrease
The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.
ln x values
ln 1 = 0
Basic formulas for natural logarithms
Formulas following from the definition of the inverse function:
The main property of logarithms and its consequences
Base replacement formula
Any logarithm can be expressed in terms of natural logarithms using the base substitution formula:
Proofs of these formulas are presented in the section "Logarithm".
Inverse function
The inverse of the natural logarithm is the exponent.
If , then
If, then.
Derivative ln x
.
Derivative of the natural logarithm:
.
Derivative of the natural logarithm of modulus x:
.
Derivative of nth order:
Deriving formulas > > >
Integral
.
The integral is calculated by integration by parts:
Expressions using complex numbers
Consider the function of the complex variable z:
.
Let's express the complex variable z via module r and argument φ
:
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If you put
, where n is an integer,
it will be the same number for different n.
Therefore, the natural logarithm, as a function of a complex variable, is not a single-valued function.
Power series expansion
When the expansion takes place:
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
One of the elements of primitive level algebra is the logarithm. The name comes from the Greek language from the word “number” or “power” and means the power to which the number in the base must be raised to find the final number.
Types of logarithms
- log a b – logarithm of the number b to base a (a > 0, a ≠ 1, b > 0);
- log b – decimal logarithm (logarithm to base 10, a = 10);
- ln b – natural logarithm (logarithm to base e, a = e).
How to solve logarithms?
The logarithm of b to base a is an exponent that requires b to be raised to base a. The result obtained is pronounced like this: “logarithm of b to base a.” The solution to logarithmic problems is that you need to determine the given power in numbers from the specified numbers. There are some basic rules to determine or solve the logarithm, as well as convert the notation itself. Using them, logarithmic equations are solved, derivatives are found, integrals are solved, and many other operations are carried out. Basically, the solution to the logarithm itself is its simplified notation. Below are the basic formulas and properties:
For any a ; a > 0; a ≠ 1 and for any x ; y > 0.
- a log a b = b – basic logarithmic identity
- log a 1 = 0
- loga a = 1
- log a (x y) = log a x + log a y
- log a x/ y = log a x – log a y
- log a 1/x = -log a x
- log a x p = p log a x
- log a k x = 1/k log a x , for k ≠ 0
- log a x = log a c x c
- log a x = log b x/ log b a – formula for moving to a new base
- log a x = 1/log x a
How to solve logarithms - step-by-step instructions for solving
- First, write down the required equation.
Please note: if the base logarithm is 10, then the entry is shortened, resulting in a decimal logarithm. If there is a natural number e, then we write it down, reducing it to a natural logarithm. This means that the result of all logarithms is the power to which the base number is raised to obtain the number b.
Directly, the solution lies in calculating this degree. Before solving an expression with a logarithm, it must be simplified according to the rule, that is, using formulas. You can find the main identities by going back a little in the article.
When adding and subtracting logarithms with two different numbers but with the same bases, replace with one logarithm with the product or division of the numbers b and c, respectively. In this case, you can apply the formula for moving to another base (see above).
If you use expressions to simplify a logarithm, there are some limitations to consider. And that is: the base of the logarithm a is only a positive number, but not equal to one. The number b, like a, must be greater than zero.
There are cases where, by simplifying an expression, you will not be able to calculate the logarithm numerically. It happens that such an expression does not make sense, because many powers are irrational numbers. Under this condition, leave the power of the number as a logarithm.
\(a^(b)=c\) \(\Leftrightarrow\) \(\log_(a)(c)=b\)
Let's explain it more simply. For example, \(\log_(2)(8)\) is equal to the power to which \(2\) must be raised to get \(8\). 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 that is a fractional power, which means the 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: \(3^(x)=8\).What is x equal to? 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 as a decimal, 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 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 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 you can write \(\log_(2)(4)\) instead of two.
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 the basic logarithmic identity.
Logarithms based on 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.
The limit of a constant value is equal to this constant value
.
The limit of the sum (difference) of a finite number of 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's 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 of a material point
. Let us find the limit of this ratio, taking into account that
.
Consequently, determining the instantaneous speed of movement of a material point is reduced to finding the derivative of the path with respect to time.
2.2. Geometric value of the 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 a 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.
Today we will talk about logarithmic formulas and we will give indicative solution examples.
They themselves imply solution patterns according to the basic properties of logarithms. Before applying logarithm formulas to solve, let us remind you of all the properties:
Now, based on these formulas (properties), we will show examples of solving logarithms.
Examples of solving logarithms based on formulas.
Logarithm a positive number b to base a (denoted by log a b) is an exponent to which a must be raised to get b, with b > 0, a > 0, and 1.
According to the definition, log a b = x, which is equivalent to a x = b, therefore log a a x = x.
Logarithms, examples:
log 2 8 = 3, because 2 3 = 8
log 7 49 = 2, because 7 2 = 49
log 5 1/5 = -1, because 5 -1 = 1/5
Decimal logarithm- this is an ordinary logarithm, the base of which is 10. It is denoted as lg.
log 10 100 = 2, because 10 2 = 100
Natural logarithm- also an ordinary logarithm, a logarithm, but with the base e (e = 2.71828... - an irrational number). Denoted as ln.
It is advisable to memorize the formulas or properties of logarithms, because we will need them later when solving logarithms, logarithmic equations and inequalities. Let's work through each formula again with examples.
- Basic logarithmic identity
a log a b = b8 2log 8 3 = (8 2log 8 3) 2 = 3 2 = 9
- The logarithm of the product is equal to the sum of the logarithms
log a (bc) = log a b + log a clog 3 8.1 + log 3 10 = log 3 (8.1*10) = log 3 81 = 4
- The logarithm of the quotient is equal to the difference of the logarithms
log a (b/c) = log a b - log a c9 log 5 50 /9 log 5 2 = 9 log 5 50- log 5 2 = 9 log 5 25 = 9 2 = 81
- Properties of the power of a logarithmic number and the base of the logarithm
Exponent of the logarithmic number log a b m = mlog a b
Exponent of the base of the logarithm log a n b =1/n*log a b
log a n b m = m/n*log a b,
if m = n, we get log a n b n = log a b
log 4 9 = log 2 2 3 2 = log 2 3
- Transition to a new foundation
log a b = log c b/log c a,if c = b, we get log b b = 1
then log a b = 1/log b a
log 0.8 3*log 3 1.25 = log 0.8 3*log 0.8 1.25/log 0.8 3 = log 0.8 1.25 = log 4/5 5/4 = -1
As you can see, the formulas for logarithms are not as complicated as they seem. Now, having looked at examples of solving logarithms, we can move on to logarithmic equations. We will look at examples of solving logarithmic equations in more detail in the article: "". Do not miss!
If you still have questions about the solution, write them in the comments to the article.
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