# Logarithmic Properties – ✅ Log Formulas ⭐️⭐️⭐️⭐️⭐️

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Logarithmic formulas are an important topic in the high school mathematics curriculum. Here are all the details about Logarithmic formulas you need to know to apply and learn well.

Logarithm abbreviated Log is the opposite of exponential. Thus, the logarithm of a number is the exponent of the base (fixed value) raised to the power to produce another number. Simply put, a logarithm is a multiplication that has a number of repetitions. Example: logsAx=y equals ay= x. If the base 10 logarithm of 1000 is 3. We have, 103is 1000 means 1000 = 10 x 10 x 10 = 103. So, the multiplication in the example is repeated 3 times.

In short, exponential allows a positive number to be raised to the power of any exponent which always results in a positive number. Therefore, logarithms are used to calculate the product of any 2 positive numbers, provided they have a positive number #1.

In order to understand and apply this logarithm formula firmly in solving math problems, you need to understand the logarithm formula and how to apply it. Here are steps to help you understand the logarithm formula thoroughly.

It’s very simple to find the difference. Logarithmic equations have the following form: logAx=y

Thus, logarithmic equations always have a log letter. If an exponential equation means that the variable is raised to a power then it is an exponential equation. The exponent is placed after the number.

Logarithm: notesAx=y

Index number: Ay= x

An example of a logarithmic formula: log28=3

Logarithmic formula components: Log stands for logarithm. The base is 2. The argument is 8. The exponent is 3.

You need to know the logarithm has many types to differentiate properly. Logarithms include:

• The decimal logarithm or base 10 logarithm is written as a log10b is usually written as lgb or logb. The base 10 logarithm has all the properties of logarithms with a base > 1. The formula: lgb=α↔10A=b

• The natural or logarithm of the base e (where e ≈ 2.718281828459045), written as a logeb number is usually written as lnb. The formula is as follows: lnb=α↔eA=b

Moreover, based on the logarithmic property, we have the following types:

• Unit logarithm and base logarithm. Therefore, with an arbitrary basis, we will always have the following logarithmic formula: logA1=0 and logAa=1

• Exponent and logarithm with the same base. Where the exponential of the real number α with base a is the calculation of aα; and the positive logarithm of B in base a will calculate the log as two opposite operations a,b>0(a≠1) anotesAA=logaaα=αaloga⁡α=logA⁡aA=a

notesABA=αlogabloga⁡bα=αlogA⁡b

Logarithms and operations

• Bases conversion enables the conversion of operations with logarithms of different bases when calculating logarithms of the same common base. With this logarithm formula, when you know the logarithm of the base α, you will be able to calculate any base just like you can calculate the logarithm of base 2, 3 based on the logarithm of base 10.

Given two positive numbers a and b with a#1 we have the following logarithmic properties:

notesA(1)=0

notesA(a)=1

AnotesAB=b

notesAAA=a

The logarithmic property helps you solve logarithmic and exponential equations. Without these properties, you will not be able to solve the equation. The logarithmic property is available only if the base and logarithmic arguments are positive, provided that the base a# 1 or 0.

• Property 1: notesA(xy)=logAx+=logayloga⁡(xy)=loga⁡x+=logA⁡y

The logarithm of two numbers x and y multiplied together can be divided into 2 separate logarithms by addition.

For example:

notes216=log2(8.2)=log28+log22=3+1=4

• Property 2: notesA(x/y)=logAx−logAy

The logarithm of two numbers x and y divided by each other can be divided into 2 logarithms by subtraction. Thus, the base x logarithm will subtract from the base y logarithm.

We have the following logarithmic formula: logABA=anotesABcondition for all numbers α and a, b is positive for #1.

Regarding logarithm formulas and quick solutions, you should be interested in the logarithm of the exponential function, the logarithm of the exponential function and the logarithm of the function. While the formulas aren’t difficult, it’s easy to mess up conditions when doing any type of math. The key to you doing well is studying the theory thoroughly, knowing exactly what the problem is will help you avoid it. Also memorize the logarithm formula by doing repeated exercises and trying different types of problems.

With a logarithm table, you will calculate much faster than a calculator, especially when you want to do quick calculations or multiply large numbers, using logarithms is more convenient.

In order to quickly find the logarithm, you need to pay attention to the following information:

• Choose the right table: Most logarithm tables are for base 10 logarithms called decimal logarithms.

• Find the correct cell: The value of the cell at the intersection of the vertical and horizontal rows.

• Find the correct number using the smaller column on the right side of the table. Use this method if the number has 4 or more.

• Finding the prefix before the decimal: Logarithmic tables tell you the prefix before the decimal. The part after the comma is called the mantissa.

• Find the integer part. This method is easiest to find for base 10 logarithms. You find it by counting the remaining decimal digits and subtracting one digit.

To solve advanced logarithmic equations, you need to remember the following:

• Understand what logarithm is? For example 10^2 is 100, 10^3 is 1000. So the exponent 2,3 is the base 10 logarithm of 100 and 1000. Each logarithm table can only be used with a certain base. By far the most common type of log table is the base 10 logarithm, also known as the common logarithm.

• Determine the property of the number whose logarithm you want to find

• When searching the logarithm table, you should carefully look at the leftmost vertical row with your finger to calculate the logarithm in the table. Then you slide your finger to find the intersection between the vertical and horizontal rows.

• If the logarithm table has a small subtable that is used for large calculations or you want to find a more precise value, slide your finger to the column in the table marked with the next digit of the number you are looking for.

• Add the numbers found in the previous 2 steps.

• Add the feature: When you find the point where two lines intersect to find the number you want to find, add the feature to the mantissa above to get your logarithmic result.

To master knowledge related to Logarithms, you can apply the following 6 methods:

Book contents:
Topic 1. Hats – Logarithmic
Problem 1. Exponential – Exponential – Logarithmic
+ Topic 1. Exponential – Logarithmic
+ Topic 2. Exponential and logarithmic functions
Problem 2. Exponential and logarithmic equations
Problem 3. Exponential and logarithmic inequalities
2. Exponential and logarithmic methods
3. Sub-hidden method
4. Solve exponential-logarithmic inequalities using the function method
5. Solve exponential – logarithmic inequalities using the evaluation – inequality method
Problem 4. Systems of equations and systems of exponential – logarithmic inequalities
+ Form 1. Solve the exponential-logarithmic system by the equality transformation method
+ Form 2. Solve the exponential logarithmic system by setting the sub-unknowns
+ Form 3. Solve the exponential – logarithmic system by the function method
+ Form 4. Solve exponential – logarithmic systems using the inequality evaluation method
Topic 2. Complex numbers
Question 1. Complex numbers
Problem 2. The problem of the geometric representation of complex numbers
Problem 3. Find the complex number with the largest and smallest modulus
Problem 4. Square roots of complex numbers and square root equations – Equations reduced to squares – Systems of equations
Problem 5. Trigonometry forms of complex numbers

Shared source: Kien Giang College (Kgtec.edu.vn)

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## All the details about the LOGARITE formula you need to know

Logarithmic formulas are an important topic in the high school mathematics curriculum. Here are all the details about Logarithmic formulas you need to know to apply and learn well.

Logarithm abbreviated Log is the opposite of exponential. Thus, the logarithm of a number is the exponent of the base (fixed value) raised to the power to produce another number. Simply put, a logarithm is a multiplication that has a number of repetitions. Example: logsAx=y equals ay= x. If the base 10 logarithm of 1000 is 3. We have, 103is 1000 means 1000 = 10 x 10 x 10 = 103. So, the multiplication in the example is repeated 3 times.

In short, exponential allows a positive number to be raised to the power of any exponent which always results in a positive number. Therefore, logarithms are used to calculate the product of any 2 positive numbers, provided they have a positive number #1.

In order to understand and apply this logarithm formula firmly in solving math problems, you need to understand the logarithm formula and how to apply it. Here are steps to help you understand the logarithm formula thoroughly.

It’s very simple to find the difference. Logarithmic equations have the following form: logAx=y

Thus, logarithmic equations always have a log letter. If an exponential equation means that the variable is raised to a power then it is an exponential equation. The exponent is placed after the number.

Logarithm: notesAx=y

Index number: Ay= x

An example of a logarithmic formula: log28=3

Logarithmic formula components: Log stands for logarithm. The base is 2. The argument is 8. The exponent is 3.

You need to know the logarithm has many types to differentiate properly. Logarithms include:

• The decimal logarithm or base 10 logarithm is written as a log10b is usually written as lgb or logb. The base 10 logarithm has all the properties of logarithms with a base > 1. The formula: lgb=α↔10A=b

• The natural or logarithm of the base e (where e ≈ 2.718281828459045), written as a logeb number is usually written as lnb. The formula is as follows: lnb=α↔eA=b

Moreover, based on the logarithmic property, we have the following types:

• Unit logarithm and base logarithm. Therefore, with an arbitrary basis, we will always have the following logarithmic formula: logA1=0 and logAa=1

• Exponent and logarithm with the same base. Where the exponential of the real number α with base a is the calculation of aα; and the positive logarithm of B in base a will calculate the log as two opposite operations a,b>0(a≠1) anotesAA=logaaα=αaloga⁡α=logA⁡aA=a

notesABA=αlogabloga⁡bα=αlogA⁡b

Logarithms and operations

• Bases conversion enables the conversion of operations with logarithms of different bases when calculating logarithms of the same common base. With this logarithm formula, when you know the logarithm of the base α, you will be able to calculate any base just like you can calculate the logarithm of base 2, 3 based on the logarithm of base 10.

Given two positive numbers a and b with a#1 we have the following logarithmic properties:

notesA(1)=0

notesA(a)=1

AnotesAB=b

notesAAA=a

The logarithmic property helps you solve logarithmic and exponential equations. Without these properties, you will not be able to solve the equation. The logarithmic property is available only if the base and logarithmic arguments are positive, provided that the base a# 1 or 0.

• Property 1: notesA(xy)=logAx+=logayloga⁡(xy)=loga⁡x+=logA⁡y

The logarithm of two numbers x and y multiplied together can be divided into 2 separate logarithms by addition.

For example:

notes216=log2(8.2)=log28+log22=3+1=4

• Property 2: notesA(x/y)=logAx−logAy

The logarithm of two numbers x and y divided by each other can be divided into 2 logarithms by subtraction. Thus, the base x logarithm will subtract from the base y logarithm.

We have the following logarithmic formula: logABA=anotesABcondition for all numbers α and a, b is positive for #1.

Regarding logarithm formulas and quick solutions, you should be interested in the logarithm of the exponential function, the logarithm of the exponential function and the logarithm of the function. While the formulas aren’t difficult, it’s easy to mess up conditions when doing any type of math. The key to you doing well is studying the theory thoroughly, knowing exactly what the problem is will help you avoid it. Also memorize the logarithm formula by doing repeated exercises and trying different types of problems.

With a logarithm table, you will calculate much faster than a calculator, especially when you want to do quick calculations or multiply large numbers, using logarithms is more convenient.

In order to quickly find the logarithm, you need to pay attention to the following information:

• Choose the right table: Most logarithm tables are for base 10 logarithms called decimal logarithms.

• Find the correct cell: The value of the cell at the intersection of the vertical and horizontal rows.

• Find the correct number using the smaller column on the right side of the table. Use this method if the number has 4 or more.

• Finding the prefix before the decimal: Logarithmic tables tell you the prefix before the decimal. The part after the comma is called the mantissa.

• Find the integer part. This method is easiest to find for base 10 logarithms. You find it by counting the remaining decimal digits and subtracting one digit.

To solve advanced logarithmic equations, you need to remember the following:

• Understand what logarithm is? For example 10^2 is 100, 10^3 is 1000. So the exponent 2,3 is the base 10 logarithm of 100 and 1000. Each logarithm table can only be used with a certain base. By far the most common type of log table is the base 10 logarithm, also known as the common logarithm.

• Determine the property of the number whose logarithm you want to find

• When searching the logarithm table, you should carefully look at the leftmost vertical row with your finger to calculate the logarithm in the table. Then you slide your finger to find the intersection between the vertical and horizontal rows.

• If the logarithm table has a small subtable that is used for large calculations or you want to find a more precise value, slide your finger to the column in the table marked with the next digit of the number you are looking for.

• Add the numbers found in the previous 2 steps.

• Add the feature: When you find the point where two lines intersect to find the number you want to find, add the feature to the mantissa above to get your logarithmic result.

To master knowledge related to Logarithms, you can apply the following 6 methods:

Book contents:
Topic 1. Hats – Logarithmic
Problem 1. Exponential – Exponential – Logarithmic
+ Topic 1. Exponential – Logarithmic
+ Topic 2. Exponential and logarithmic functions
Problem 2. Exponential and logarithmic equations
Problem 3. Exponential and logarithmic inequalities
2. Exponential and logarithmic methods
3. Sub-hidden method
4. Solve exponential-logarithmic inequalities using the function method
5. Solve exponential – logarithmic inequalities using the evaluation – inequality method
Problem 4. Systems of equations and systems of exponential – logarithmic inequalities
+ Form 1. Solve the exponential-logarithmic system by the equality transformation method
+ Form 2. Solve the exponential logarithmic system by setting the sub-unknowns
+ Form 3. Solve the exponential – logarithmic system by the function method
+ Form 4. Solve exponential – logarithmic systems using the inequality evaluation method
Topic 2. Complex numbers
Question 1. Complex numbers
Problem 2. The problem of the geometric representation of complex numbers
Problem 3. Find the complex number with the largest and smallest modulus
Problem 4. Square roots of complex numbers and square root equations – Equations reduced to squares – Systems of equations
Problem 5. Trigonometry forms of complex numbers

## Types of questions about derivative formulas

The function y = f(x) has a derivative at point x= xf'(x )=f'(x )

A function y = f(x) that has a derivative at a point must first be continuous at that point.

Example 1: f(x) = 2x +1 at x=2 Example 1: If y = e .sinx, prove the relation y”+2y′+ 2y = 0

Solution:

We have y′=−e .sinx + e .cosx

y′ =−e .sinx+e−x.cosx

y”=e .sinx−e .cosx−e .cosx−e .sinx = −2e .cosx

Jadi y” + 2y′ + 2y = −2.e .cosx− −2.e .sinx + 2.e .cosx + 2.e .sinx =0

The equation for the tangent to the curve (C): y= f(x) at the point of contact M( x ;y ) has the form:

Example: Given the function y= x +3mx + ( m+1)x + 1 (1), m is the actual parameter. Find the values ​​of

m so that the tangent to the graph of the function (1) at the point with coordinates x = -1 passes through point A(

1;2).

The set of determinations D = R

y’ = f’(x)= 3x + 6mx + m + 1

With x = -1 => y = 2m -1, f'( -1) = -5m + 4

The equation of the tangent line at point M( -1; 2m – 1) : y= ( -5m + 4 ) ( x+1) + 2m -1 (d)

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We have A ( 1,2) (d)( -5m + 4).2 + 2m – 1 = 2 => m = 5/8

Write the PTTT of ( C ) : y = f( x ), knowing that it has a slope of k

Let M( x ;y ) be a contact. Compute y’ => y'(x )

Because the tangent equation has a slope k => y’ = ( x ) = k (i)

Solve (i) to find x => y = f(x ) => : y = k (x – x )+ y

Note: The slope k = y'( x ) of the tangent Δ is usually given indirectly as follows: Example: Given the function y=x +3x -9x+5 ( C). Of all the tangents to the graph ( C ), suppose

Find the tangent line with the smallest slope.

Ta có y’ = f’( x ) = 3x + 6x – 9

Let x be the coordinate of the tangent of the tangent line, so f'( x ) = 3 x + 6 x – 9

We have 3 x + 6 x – 9 = 3 ( x + 2x +1) – 12 = 3 (x + 1) – 12 > – 12

So mean f( x )= – 12 at x = -1 => y = 16

Derive the tangent equation to find: y= -12( x+1)+16y= -12x + 4 Check out the Marathon Education online course now

The Marathon Education team has just shared with you important knowledge about logarithmic functions and calculation formulas log derivative . We hope that this article will equip them with the necessary background knowledge to help them study Maths better and get high scores in their upcoming exams. To online online learning there’s still a lot of other content, don’t forget to follow Marathon every day. I wish you success!

## All the details about the LOGARITE formula you need to know

Logarithmic formulas are an important topic in the high school mathematics curriculum. Here are all the details about Logarithmic formulas you need to know to apply and learn well.

Logarithm abbreviated Log is the opposite of exponential. Thus, the logarithm of a number is the exponent of the base (fixed value) raised to the power to produce another number. Simply put, logarithms are multiplication repeated over and over again. Example: logsAx=y equals ay= x. if the base 10 logarithm of 1000 is 3. We have, 103is 1000 i.e. 1000 = 10 x 10 x 10 = 103. similarly, the multiplication in the example is repeated 3 times.

In short, exponential allows a positive number to be raised to the power of any exponent which always results in a positive number. Therefore, the logarithm used to calculate the product of any 2 positive numbers, the condition is that it has a positive number #1.

In order to understand and apply these logarithmic formulas powerfully to do math exercises, you need to understand the Logarithm formula and how to apply it. Here are steps to help you understand the logarithm formula thoroughly.

It’s very simple to see the difference. Logarithmic equations have the following form: logAx=y

similarly, logarithmic equations always have the log of letters. If an equation has an exponential i.e. the variable is raised to a power then it is an exponential equation. The exponent is placed after the number.

Logarithm: notesAx=y

Index number: Ay= x

An example of a logarithmic formula: log28=3

logarithmic formula components: Log stands for logarithm. The base is 2. The argument is 8. The exponent is 3.

You need to know that there are many types of logarithms to distinguish them properly. Logarithms include:

• The decimal logarithm or base 10 logarithm is written as a log10b is usually written as lgb or logb. The base 10 logarithm has all the properties of logarithms with a base > 1. The formula: lgb=α↔10A=b

• The natural or logarithm of the base e (where e ≈ 2.718281828459045), written as a logeb number is usually written as lnb. The formula is as follows: lnb=α↔eA=b

Moreover, based on the logarithmic property, we have the following types:

• Unit logarithm and base logarithm. Therefore, with an arbitrary basis, we will always have the following logarithmic formula: logA1=0 and logAa=1

• Exponent and logarithm with the same base. Where the exponential of the real number α with base a is the calculation of aα; and the positive logarithm of B in base a will calculate the log as two opposite operations a,b>0(a≠1) anotesAA=logaaα=αaloga⁡α=logA⁡aA=a

notesABA=αlogabloga⁡bα=αlogA⁡b

Logarithms and mathematical operations

• The rad conversion allows conversion of operations that take logarithms of different bases when calculating logarithms of the same common base. With this logarithm formula, when you know the logarithm of the base α, you will be able to calculate any base just like you can calculate the logarithm of base 2, 3 based on the logarithm of base 10.

Given two positive numbers a and b with a#1 we have the following logarithmic properties:

notesA(1)=0

notesA(a)=1

AnotesAB=b

notesAAA=a

The logarithmic property helps you solve logarithmic and exponential equations. Without these properties, you will not be able to solve the equation. The logarithm property is available only when the base and logarithm arguments are positive, the condition is base a #1 or 0.

• Property 1: notesA(xy)=logAx+=logayloga⁡(xy)=loga⁡x+=logA⁡y

The logarithm of two numbers x and y multiplied together can be divided into 2 separate logarithms by addition.

For example:

notes216=log2(8.2)=log28+log22=3+1=4

• Property 2: notesA(x/y)=logAx−logAy

The logarithm of two numbers x and y divided by each other can be divided into 2 logarithms by subtraction. Thus, the base x logarithm will subtract from the base y logarithm.

We have the following logarithmic formula: logABA=anotesABcondition for all numbers α and a, b is positive for #1.

Regarding logarithm formulas and quick solutions, you should be interested in the logarithm of the exponential function, the logarithm of the exponential function and the logarithm. While the formulas aren’t difficult, it’s easy to mess up conditions when doing any type of math. The key for you to do well is to study the theory well, understand the problems that will help you avoid them. Also memorize the logarithm formula by doing repeated exercises and trying different types of problems.

With a logarithm table, you will calculate much faster than a calculator, especially when you want to do quick calculations or multiply large numbers, using logarithms is more convenient.

In order to quickly find the logarithm, you need to pay attention to the following information:

• choose the right table: Most logarithm tables are for base 10 logarithms called decimal logarithms.

• Find the correct cell: the cell value at the intersection of the vertical and horizontal rows.

• Find the most correct number using the smaller column on the right side of the table. Use this method when numbers have 4 or more.

• Finding the prefix before the decimal: Logarithmic tables tell you the prefix before the decimal. The part after the comma is called the mantissa.

• Find the integer part. This method is easiest to find for base 10 logarithms. You find it by counting the remaining decimal digits and subtracting one digit.

To solve increasing logarithmic equations, you need to remember the following:

• Understand what logarithm is? For example, 10^2 is 100, 10^3 is 1000. Similarly, the exponent 2,3 is the base 10 logarithm of 100 and 1000. Each logarithm table can only be used with a certain base. By far the most common type of log table is the base 10 logarithm, also known as the common logarithm.

• Determine the property of the number whose logarithm you want to find

• When searching the logarithm table, use your finger carefully to find the leftmost vertical row to calculate the logarithm in the table. Then you slide your finger to find the intersection between the vertical and horizontal rows.

• If the logarithm table has small subtables that are used to calculate large calculations or want to find a more precise value, slide your finger to the column in the table marked with the next digit of the number you are looking for.

• Add the numbers found in the previous 2 steps.

• Add the feature: When you find the point where two lines intersect to find the number you want to find, add the feature to the mantissa above to get your logarithmic result.

To master knowledge related to Logarithms, you can apply the following 6 ways:

Book contents:
Topic 1. Hats – Logarithmic
Problem 1. Exponential – Exponential – Logarithmic
+ Topic 1. Exponential – Logarithmic
+ Topic 2. Exponential and logarithmic functions
Problem 2. Exponential and logarithmic equations
Problem 3. Exponential and logarithmic inequalities
2. Exponential and logarithmic methods
3. Sub-hidden method
4. Solve exponential-logarithmic inequalities using the function method
5. Solve exponential – logarithmic inequalities using the evaluation – inequality method
Problem 4. Systems of equations and systems of exponential – logarithmic inequalities
+ Form 1. Solve the exponential-logarithmic system by the equality transformation method
+ Form 2. Solve the exponential logarithmic system by setting the sub-unknowns
+ Form 3. Solve the exponential – logarithmic system by the function method
+ Form 4. Solve exponential – logarithmic systems using the inequality evaluation method
Topic 2. Complex numbers
Question 1. Complex numbers
Problem 2. Problem geometric representation of complex numbers
Problem 3. Find the complex number with the largest and smallest modulus
Problem 4. Square roots of complex numbers and square root equations – equations reduced to squares – Systems of equations
Problem 5. Trigonometry forms of complex numbers

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## All the details about the LOGARITE formula you need to know

Logarithmic formulas are an important topic in the high school mathematics curriculum. Here are all the details about Logarithmic formulas you need to know to apply and learn well.

You’re viewing: Logarithmic Properties

Logarithm abbreviated Log is the opposite of exponential. Thus, the logarithm of a number is the exponent of the base (fixed value) raised to the power to produce another number. Simply put, a logarithm is a multiplication that has a number of repetitions. Example: logax=y is the same as ay=x. If the base 10 logarithm of 1000 is 3. We have, 103 is 1000 which means 1000 = 10 x 10 x 10 = 103. So, the multiplication in the example is repeated 3 times.

In short, exponential allows a positive number to be raised to the power of any exponent which always results in a positive number. Therefore, logarithms are used to calculate the product of any 2 positive numbers, provided they have a positive number #1.

In order to understand and apply this logarithm formula firmly in solving math problems, you need to understand the logarithm formula and how to apply it. Here are steps to help you understand the logarithm formula thoroughly.

It’s very simple to find the difference. The logarithmic equation looks like this: logax=y

Thus, logarithmic equations always have a log letter. If an exponential equation means that the variable is raised to a power then it is an exponential equation. The exponent is placed after the number.

Logarithm: logax=y

Index number: is = x

An example of a logarithmic formula: log 28=3

Logarithmic formula components: Log stands for logarithm. The base is 2. The argument is 8. The exponent is 3.

You need to know the logarithm has many types to differentiate properly. Logarithms include:

• Decimal logarithms or base 10 logarithms are written log10b, usually written lgb or logb. The base 10 logarithm has all the properties of logarithms with a base > 1. The formula: lgb=α↔10α=b

•The natural or base logarithm of e (where e ≈ 2.718281828459045), written as a logeb number is usually written as lnb. The formula is as follows: lnb=α↔eα=b

Moreover, based on the logarithmic property, we have the following types:

• Unit logarithm and base logarithm. Thus, with any radix, we will always have the following logarithmic formula: loga1=0 and logaa=1

• Exponential and logarithmic to the same base. Where the exponential of the real number α with base a is the calculation of aα; and the positive digital logarithm of B according to base a will calculate the logab as two opposite operations ∀a,b>0(a≠1)alogaα=logaaα=αaloga⁡α=loga⁡aα=α

logabα=αlogabloga⁡bα=αloga⁡b

Logarithms and operations

• Bases conversion enables the conversion of operations with logarithms of different bases when calculating logarithms of the same common base. With this logarithm formula, when you know the logarithm of the base α, you will be able to calculate any base just like you can calculate the logarithm of base 2, 3 based on the logarithm of base 10.

Given two positive numbers a and b with a#1 we have the following logarithmic properties:

window(1)=0

log(a)=1

alogab=b

logaaaa=a

The logarithmic property helps you solve logarithmic and exponential equations. Without these properties, you will not be able to solve the equation. The logarithmic property is available only if the base and logarithmic arguments are positive, provided that the base a# 1 or 0.

• Property 1: loga(xy)=logax+=logayloga⁡(xy)=loga⁡x+=loga⁡y

The logarithm of two numbers x and y multiplied together can be divided into 2 separate logarithms by addition.

For example:

log216=log2(8.2)=log28+log22=3+1=4

• Property 2: loga(x/y)=logax−logay

The logarithm of two numbers x and y divided by each other can be divided into 2 logarithms by subtraction. Thus, the base x logarithm will subtract from the base y logarithm.

We have the following logarithmic formula: logabα=αlogab condition for all numbers α and a, b is positive for a #1.

Regarding logarithm formulas and quick solutions, you should be interested in the logarithm of the exponential function, the logarithm of the exponential function and the logarithm of the function. While the formulas aren’t difficult, it’s easy to mess up conditions when doing any type of math. The key to you doing well is studying the theory thoroughly, knowing exactly what the problem is will help you avoid it. Also memorize the logarithm formula by doing repeated exercises and trying different types of problems.

With a logarithm table, you will calculate much faster than a calculator, especially when you want to do quick calculations or multiply large numbers, using logarithms is more convenient.

In order to quickly find the logarithm, you need to pay attention to the following information:

• Choose the right table: Most logarithm tables are for base 10 logarithms called decimal logarithms.

• Find the correct cell: The value of the cell at the intersection of the vertical and horizontal rows.

• Find the correct number using the smaller column on the right side of the table. Use this method if the number has 4 or more.

• Find the prefix before the decimal: Logarithmic tables tell you the prefix before the decimal. The part after the comma is called the mantissa.

• Find the integer part. This method is easiest to find for base 10 logarithms. You find it by counting the remaining decimal digits and subtracting one digit.

To solve advanced logarithmic equations, you need to remember the following:

• Understand what logarithms are? For example 10^2 is 100, 10^3 is 1000. So the exponent 2,3 is the base 10 logarithm of 100 and 1000. Each logarithm table can only be used with a certain base. By far the most common type of log table is the base 10 logarithm, also known as the common logarithm.

• Define the property of the number for which you want to find the logarithm

•When searching the logarithm table, you should carefully use your finger to find the leftmost vertical row to calculate the logarithm in the table. Then you slide your finger to find the intersection between the vertical and horizontal rows.

•If the logarithm table has a small sub-table for large calculations or you want to find a more precise value, slide your finger to the column in the table marked with the next digit of the number you are looking for.

• Add up the numbers found in the previous 2 steps.

See also: Top 19 articles analyzing Chi Pheo’s process of decay, Analyzing Chi Pheo’s process of alienation

• Add the feature: When you find the point where two lines intersect to find the number you want to find, add the feature to the mantissa above to get your logarithmic result.

To master knowledge related to Logarithms, you can apply the following 6 methods:

Contents of the book: Topic 1. Exponential – Logarithmic Problems 1. Exponential – Exponential – Logarithmic + Topic 1. Exponential – Logarithmic + Problems 2. Exponential and Logarithmic Function Problems 2. Exponential and Logarithmic Equations Problems 3. Exponential and logarithmic inequalities 1. The method returns the same base2. Exponential and logarithmic methods3. Sub-hidden method 4. Solve logarithmic-exponential inequalities with the functional method5. Solving exponential – logarithmic inequalities using the evaluation – inequality method Problem 4. Systems of equations and systems of exponential inequalities – logarithmic+ Form 1. Solving exponential – logarithmic systems using the equality transformation method+ Form 2. Solving exponential – logarithmic systems by assigning unknown minor numbers+ Form 3 Solving exponential – logarithmic systems using functional method + Form 4. Solving exponential – logarithmic systems using inequalities Evaluation Theme 2. Complex numbers Problem 1. Complex numbers Problem 2. Problems with geometric representation of complex numbers Problem 3. Finding complex numbers with the largest and smallest modulus Problem 4. Square roots of complex numbers and square root equations – Equations reduced to the second degree – Systems of equations Problem 5. Trigonometric forms of complex numbers

Reference source:

• Log formula – Kien Giang College,  https://kgtec.edu.vn/cong-thuc-log
• Log formula – Kindergarten Secondary Education – Kindergarten Hanoi,  https://tcspmgnthn.edu.vn/cong-thuc-log/
• Log derivative synthesis, logarithmic, square root, x root, trigonometric formulas,   https://blog.marathon.edu.vn/dao-ham-log/
• Log formula – Cmm.edu.vn | Center for Natural Resources and Environment College,  https://cmm.edu.vn/cong-thuc-log-cmm-edu-vn.html
• Logarithmic Properties – ✅ Log Formulas ⭐️⭐️⭐️⭐️⭐️,  https://magdalenarybarikova.com/tinh-chat-logarithmic/

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