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### Introduction to logarithms

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__Applied Mathematics__

__Applied Mathematics__

__Introduction to logarithms__

__Introduction to logarithms__

**Simple explanations and the implementations of logarithmic formulas in solving complicated mathematical problems. Indices, logarithmic formulas and logarithmic tables are explained here**

**Logarithmic Tables**

**********************

__Learning Objectives of Teaching & Learning Logarithms__

__Learning Objectives of Teaching & Learning Logarithms__

**Students should study the logarithms to achieve the following objectives:- **

**Understanding the concept of
logarithms: **

The primary objective is to
help students understand what logarithms are and how they relate to
exponentiation.

**Solving logarithmic
equations: **

Students should learn how to
solve logarithmic equations, including properties of logarithms such as the
product, quotient, and power rules.

**Applying logarithms to
real-world problems: **

Students should be able to
apply logarithms to solve real-world problems, such as exponential growth and
decay, compound interest, and logarithmic scales.

__Learning
Outcomes of Teaching & Learning Logarithms__

__Learning Outcomes of Teaching & Learning Logarithms__

**After study of this topic students should
possess the following skills**

**Computational skills: **

Students should be able to perform calculations involving
logarithms, such as evaluating logarithmic expressions, simplifying logarithmic
equations, and solving logarithmic equations for unknowns.

**Problem-solving skills:**

Students should be able to apply logarithms to solve a variety of mathematical and real-world problems, including those involving exponential growth, decay, and rates of change.

**Critical thinking and reasoning:**

Students should develop critical thinking skills by analyzing and interpreting logarithmic expressions and equations in different contexts, and making connections between logarithms and exponentiation.

__Indices__**x ^{m}
= x.x.x.x…………….m terms**

**Here x is
called the base and m is called the index or exponent of x.**

**Indices is
the plural of index.**

__Laws of Indices__**(i) **** **

**(ii) **

**(iii) **

**(iv)**

**(v) **

** (vi) **

**(vii) **

**(vii) **

__Logarithms__**Logarithm of a number to a given base is the index or power to which the base must be raised so as to be equal to the given number.**

**If a ^{x} = N (Where a is a positive number ≠1), then x is called the logarithm of N to the base a and is written as x = Log_{a} N.**

**For Example:**

**Question:**

**Solution:**

**Fundamental Properties of Logarithms**

**Fundamental Properties of Logarithms**

**Product Rule:**

**Quotient Rule:**

**Power Rule:**

**Question:**** **

**Solution:**

**Natural Logarithms**

**Logarithms to the base e (= 2.7183) are called natural logarithms. They are used in all theoretical calculations. For Example : Log**

_{e }(sin x) This logarithm is called the natural logarithm.

Common Logarithms

Logarithms to the base 10 are called common logarithms. They are used in arithmetic calculations. For Example : Log

Common Logarithms

Logarithms to the base 10 are called common logarithms. They are used in arithmetic calculations. For Example : Log

_{10 }(34.56) This logarithm is called the common logarithm.

**Characteristic and Mantissa**

**The integral part of the logarithm of a number, after expressing the decimal part as positive (if not already so), is called the characteristic.**

**Mantissa**

Positive decimal part is called Mantissa. Note: Mantissa is always positive.

Positive decimal part is called Mantissa. Note: Mantissa is always positive.

Rule 1

**Rule to find the Characteristic**

Rule 1**Characteristic of the logarithm of a number > 1 is positive****Characteristic of the logarithm of a number is one less than the number of digits in the integral part of the number.**

**For Example : 34.586**

**Integral Part Before Decimal = 34****No. Digits in the integral part = 2****Characteristic = 2 - 1 = 1**

**Rule II**

**Characteristic of the logarithm of a number < 1 is negative.****It is numerically one more than the number of zeroes immediately after the decimal point.**

**For Example : 0.0586 **

**Here given number is < 1 so characteristic should be negative**

**Number of zeroes after decimal = 1**

**Characteristic = - (1 + 1) = - 2**

**Negative character always written in bar form as given below**

**-2 is written in the form of bar as**

**Characteristic and Mantissa for a negative number**

**For Example:**

**Let the number is - 2.305**

**- 2.305 = - (2 + 0. 305) = - 2 - 0.305**

**To make decimal part positive we have to subtract and add 1 to this number**

**- 2 - 1 + 1 - 0.305 = - 3 + 0.695**

**Question: Find the number of digits in 3**

^{62}**Solution**

**Let x
= 3 ^{62}**

**Taking log on both side**

**Log x = Log 3 ^{62}**

**Log x = 62 Log 3 = 62 X 0.4771 = 29.5802**

**Characteristic
= 29**

**Number of digits in 3 ^{62} = 29 + 1 = 30**

**Question: Find the position of the first significant figure in 3**

^{-65}**Solution**

**Let x = 3 ^{-65} **

**Log x = Log 3 ^{-65} = - 65 X Log 3
= -65 X 0.4771 = -31.0115**

**Log x = -(31 + 0.0115) = -31 – 0.0115**

**In order to make Mantissa positive , add and subtract 1**

**Log x =
- 31 – 1 + 1 -0.0115**

**
= - 32 + 0.9985**

**Characteristic Value = - 32**

**So there are 32 – 1= 31 zeroes immediate after
decimal.**

**So position of the first significant figure
is 31 + 1 = 32**

**Question: Find the 5th root of 0.003**

**Solution:**

**Taking ****antilog**** on both sides we get**

**Question: Find the value of**

**Log**

_{43 }57**Solution:**

**Taking antilog on both sides we get**

**x = Antilog(0.0312) = 1.074**

**Evaluate the following**

**Solution :**

**Taking log on both side we get**

**Taking Antilog on both side we get**

**THANKS FOR YOUR READING**

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