Most of the competitive entrance exams contain at least one question from the topic "Logarithms", but the aspirants are really scared of this topic. Their basic strategy on this area is simply skipping the question just because of the lack of confidence to approach the question due to the un-awareness. A basic understanding on the concept and rules of Logarithms help the aspirant to answer the direct as well as the indirect application type questions from this area.

Here we are discussing the basic fundamentals and some of the extended practical application of the concept of logarithm.

**Objectives:**

- Basic concept of logarithm.
- Properties (rules) of logarithm.
- Characteristic and Mantissa.
- Application of logarithms in quantitative problems.

## Funda 1: Basic concept of Logarithms

^{3}(ie. 8 is expressed in terms of 2 and 3)

_{2}8 and it is read as “log 8 to the base 2”

^{x}= N, then x = log

_{a}N

**Natural Logarithms (Napierian Logarithm)**

Logarithms expressed to the base "e" are called "Natural logarithms". "e" is an irrational number and approximately equal to 2.718281828………..

**Common Logarithms**

Logarithms expressed to the base "10" are called "Common Logarithms". Most of the questions from this topic are based on common logarithms. If a logarithmic expression without any base, then by default the base is "10".

## Funda 2: Properties (Rules) of Logarithms

The below furnished important properties of logarithm will help you to simplify most of the complicated logarithmic functions.

Rules | Illustration |
---|---|

log_{a}a = 1Where a > 0 and a ≠ 1 |
5^{a} = 51 = log _{5}5 |

log_{a}1 = 0Where a > 0 and a ≠ 1 |
5^{0} = 1log _{5}1=0 |

log_{a}x^{x} = xWhere x > 0, and a > 0 and a ≠ 1 Ie. log _{a}x^{x} is the inverse of the function a^{x} |
5^{2} =5^{2}2 = log _{5}5^{2 } |

a^{logaN }= N |
Let x = logaN ax = N a ^{logaN }=a^{N}= N |

Log_{a}(MN) = log_{a}M + log_{a}N |
2^{3} = 8Log _{s}8 = log_{2}(4 × 2)= log _{2}4 + log_{2}2= 2 + 1 = 3 |

Log_{a}(M/N) = log_{a}M - log_{a}N |
2^{3} = 8Log _{2}8 = log_{2}(16 ÷ 2) = log _{2}16 - log_{2}2= 4 - 1 = 3 |

log_{a}N^{p} = p log_{a}N |
4^{3} = 64 Log_{4}64 = log_{4} (4^{3}) = 3 log_{4}4 = 3 |

log_{a}b = 1/log_{b/sub>a} |
2^{3}=8^8^{1/3}=2log _{2}8 = 1/log_{8}2 = 1/1/3 =3 |

log_{a}N = log_{b}N/log_{b}a |
2^{3}=8log _{2}8 = log_{10}8/log_{10}23 log _{10}2/log_{10}2=3 |

log_{aq}=p/q log_{a}b |
16^{1}=16ie.log _{16}16=1log _{24}(4^{2}) = 2/4 log_{2}42/4 * 2 =1 |

a^{log10b}=b^{log10a} |
2^{log103}=3^{log102}2 ^{0.4771} = 3^{0.3010} ≈1.39log _{10}3=0.4771^log_{10}2=.3010 |

**Additional Property: I**

If log_{a}b = N where a, b > 0 and a ≠ 1, N is any real number, then;

- log
_{ 1/a}b = -N - log
_{a }1/b = N - log
_{ 1/a}1/b = N

**Additional Property: II**

If "x" is a real number and |X| < 1

- log(1+x) = x - x
^{2}/2 + x^{3}/3 -x^{4}/4 + ... - log(1-x) = -x - x
^{2}/2 - x^{3}/3 -x^{4}/4 - ... - log[(1+x)/(1-x)] =2[ x + x
^{3}/3 + x^{5}/5 + ...] - log2 = log(1+1) = 1 - 1/2 + 1/3 - 1/4 + ....

**Important values to remember:**

log_{10}3 = 0.4771

log_{10}2 = 0.3010

## Funda 3: Characteristic and Mantissa of common logarithms

**Characteristic:**

The integral part of a logarithm is called "Characteristic". Characteristic of "log N" is depends up on the number of integral digits in N. Eg: For finding log 314, first of all convert 314 in the mathematically standard form.

ie. 314 = 3.14 × 10^{2}

In the above standard representation, the exponent of 10 (ie 2) is the characteristic of log 314.

The characteristic of log 314 is 2.

Similarly the characteristic of log 1234 is 3, because 1234 = 1.234 × 10^{3}

And the characteristic of log 0.012 is – 2 , because 0.012 = 1.2 × 10^{- 2}

In general:

The characteristic of a logarithm can be found easily in the following way.

In log N;

**If N has "n" number of integral digits (ie. the digits on the left side of the decimal point) then the characteristic of log N is "n – 1".**

ie. Characteristic of log 102 is 2.

Characteristic of log 11.2 is 1.

Characteristic of log 1.25 is 0

**If N doesn't have any integral part then the Characteristic of log N depends up on the number of zeros immediately succeeding the decimal point. If there are "n" zeros immediately after the decimal point in N, then the Characteristic of log N is n + 1**

ie. Characteristic of log 0.005 is 3

Characteristic of log 0.02 is 2

Characteristic of log 0.432 is 1

**Mantissa:**

The decimal part of a logarithm is called "mantissa".

Eg: log 2 = 0.3010, here the mantissa is 0.3010 and characteristic is 1

log 200 = 2. 3010, here the mantissa is 0.3010 and characteristic is 2.

log 0.002 = 3, .3010, here the mantissa is 0.3010 and characteristic is 3

From the above set of examples, it is easy to understand that, there were no changes in the mantissa part of the log value.

**ie. The mantissas are same for all logarithms which have the same significant digits in the same order.**

**Conversion from positive mantissa to negative mantissa**

Consider log 0.0235 = 2 . 3710

In this value, characteristic is 2, , ie. The characteristic is a negative value, but the mantissa 0.3710 is positive.

ie, 2 = -2 however 2 3710 ≠ - 2.3710.

For converting the mantissa also as negative, the following method can apply.

2 . 3710 = - 2 + 0.3710.

= - 1 + ( -1 + 0.3710)

= - 1 + ( - 0. 6290)

= -1.6290

ie. 2 . 3710 = -1.6290

**More understanding on Characteristic**

If log N = x and the value of x is available then it is easy to make some logical conclusions about the number of integral digits or the number of zeros immediately after the decimal point in N.

Examples:

1) log 2 = 0.3010

Here N = 2

Characteristic = 0

The number of integral digits in N = 0 + 1 = 1

2) log 200 = 2.3010

Here N = 200

Characteristic = 2

The number of integral digits in N = 2 + 1 = 3

3) log 12345 = 4.0915

Here N = 12345

Characteristic = 4

The number of integral digits in N = 4 + 1 = 5

4) log 0.2 = 1.3010

Here N = 0.2

Characteristic = 1

The number of zeros after the decimal and before the first significant digit in N

= |-1 + 1| = 0

5) log 0.002 = 3.3010

Here N = 0.002

Characteristic = 3

The number of zeros after the decimal and before the first significant digit in N

=|-3+1| = 2

6) log 0.000056 = 5.7482

Here N = 0.000056

Characteristic = 5

The number of zeros after the decimal and before the first significant digit in N

= |-5+1| = 4