In this session we know about basic concepts of logarithms like definition of logarithm with examples, common logarithms, natural logarithms and properties of logarithms.

## Log Definition and Properties | logarithm tutorial | Exercise – 1

Contents

**What is Logarithm**

Take a relation between **x, p** and** y** is** x = p ^{y}**

In the above relation, we are unable to decide the value of** “y”** for a given value of “**x”** because the relation may be the relative change does not follow the criteria of ratio or difference.

For example if you wont to get **x = 25** in **x = 5 ^{y},** What should be the value of

**y**?

Here power to which **5** must be raised to get **25 **(i. e 5 x 5 = 25 & y = 2 )

Now defined the relation between “**y”** and** “x”.**

Here in **25 = 5**^{y }

**“y”** is called *logarithm* of **25** to the base **5** and it can be shortly written as **log _{5 }25 = 2
**

**Definition of Logarithm**

Let “p” be a positive real number, p ≠ 0 and p^{y} = x. Then “y” is called the logarithm of “x” to the base “p” and is written as

**log _{p }x = y , and conversely if log _{p }y = x then p^{x} = y**

In *logarithm* equation of **log _{p }x = y,** we can say that x >0 and p >o

( Since Logarithm to a negative base is not taken)

**y = log _{p }x ** is called the

*logarithm form*and

**p**is called the

^{y}= x*exponential form*of the equation connecting

**p , y & x**.

#### Some examples of exponential forms and their logarithmic forms

**1 .** Exponential form – ** a ^{y} = N**

Logarithmic form –** log _{a }N = y**

**2 .** Exponential form – ** 2 ^{4} = 16**

Logarithmic form –** log _{2 }16 = 4
**

**3 .** Exponential form – ** 10 ^{-2} = 0.01**

Logarithmic form –** log _{10 }0.01 = -2 = **

**4 .** Exponential form – ** 3 ^{-3} = 1/27**

Logarithmic form –** log _{3 }(1/27) = -3 = **

#### Some Examples on logarithms

**Example – 1 **: Find how many 3s required to multiply for getting 27?

**Answer: ** To get 27 , we have to multiply 3 of the 3s to get 27 ( i. e 3 x 3 x 3 = 27 )

So the** logarithm is 3**

Now we have to multiply 3 times of the number 3 to get 27 is 3″ i.e log_{3 }(27) = 3 &

read it as “logarithm of 27 to the base 3 is 3” or “log base 3 of 27 is 3”

**Example – 2 :** Find the value of** log _{4} 1024**

**Answer:** Here we have to multiply 5 times of the number 5 to get 1024 ( i.e 4 x 4 x 4 x 4 x 4 = 1024 )

So “logarithm of 1024 to the base 4 is 5 ” or “log base 4 of 1024 is 5

**i.e log _{4} 1024 = 5**

**Example – 3 :** Find the value of** log _{81 }(3)
**

**Answer:** Let ** log _{81 }(3)** = x

Exponential form – **81 ^{x} = 3 ⇔ (3 ^{4}) ^{x} = ( 3 ) ^{1}**

No According to exponential rules 4x = 1 **⇒** x = 1/4

**Example – 4 :** Find the value of** log _{2/3 }(8/27)**

**Answer:** Let **log _{2/3 }(8/27)** = x

Exponential form – **(2/3) ^{x} = 8/27 ⇔ (2/3) ^{x} = ( 2/3 ) ^{3}**

No According to exponential rules x = 3

**Characteristic and Mantissa**

The logarithm of a number consists of two parts – One is integral part and another is decimal part.

The integral part of the logarithm of a number is called its **characteristic** and the decimal part is called **mantissa**

For example** log _{10} 25 = 1.3979**

Here, *Characteristic* =** 1** & *Mantissa* =** 0.3979**

**Note:** Mantissa is always written as positive number.

##### Rule for write Mantissa and Characteristic:

To make the **mantissa** positive ( In case the value of the logarithm of a number is negative), subtract **1** from the integral part and add to the decimal part.

For example** log _{10} (0.5) = – 0.3010**

Thus **–** 0.3010 = **–** ( 0 + 0.3010 ) = **–** 0 **–** 0.3010 = **–** 0 **–** 1 **+** ( 1** –** 0.3010) = **–** 1 **+** 0.699

So here *mantissa* is **0.699**

When the **characteristic** is negative , it is represented by putting a** bar** on the number.

Thus in the above example instead of **-1**, we write

**For number of log base 10**

If the number of digits in a number is** ” n”** then the *characteristic* of logarithm of the number is **( n- 1)**

If *characteristic* of logarithm of a number is “** n” **then the number of digits in the number is **( n+1)**

**Note:**

**1. **The characteristic of common logarithms of any positive number less than 1 is negative.

**2.** The characteristic of common logarithms of any positive number greater than 1 is positive.

**3**. If the logarithm to any base** ‘ a ‘** given the characteristic** ‘n ‘,** then we can say that the number of integers possible is given by a ** ^{n+1}** – a

^{n}**Common Logarithms**

Logarithm to the base **“10”** are called *common logarithm*. It is denoted as **log _{10} x** simply denoted as

**log x**

i. e while a logarithm is written** without a base** than it mean base is really **10**

### Natural Logarithm

Logarithm to the base **” e”** are called *natural logarithm*. It is denoted as **log _{e} x**

Here **“e”** is a constant, which is an *irrational number* with an infinite, non-terminating value of **e = 2.718**

### Properties of Logarithm

**1**. **log _{a} 1 = 0 for a > 0 , a ≠ 1 ( i.e Log 1 to any base is Zero)
**

**Proof** : Let **log _{a} 1 = x . **Then according to

*logarithm definition*

**a ^{x} = 1** , so it is possible only if

**x = 0**

Therefore **a ^{x} = a ^{0 }⇔ x = 0. **

Hence **log _{a} 1 = 0** for all a > 0 ,

**a ≠ 1**

**2**. **log _{a} a = 0 for a > 0 , a ≠ 1 ( i.e Log of a number to the same base is 1)**

**Proof** : Let **log _{a} a = x . **Then according to

*logarithm definition*

**a ^{x} = a** , so it is possible only if

**x = 1**

Therefore **a ^{x} = a ^{1 }⇔ x = 1. **

Hence **log _{a} a = 1** for all a > 0 ,

**a ≠ 1**

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