## Logarithm - tips, tricks and questions

Logarithms make math easy. Studying this chapter will help you in solving and simplifying complicated numerical expressions. Rules of logarithm can also be used to find the square and square root of numbers.**Important terms:**

**Logarithm:**It is the power to which a number must be raised in order to get some other number.

**Common logarithm:**Logarithms with base 10 are common logarithms.

Common logarithms are written as log

_{10}x and if any expression is not indicated with a base, then base 10 is considered.

**Natural logarithm:**Logarithms with base e are natural logarithms.

Natural logarithms are written as log

_{e}x and denoted as ln (x).

**Logarithm of a number contains 2 parts:**Characteristic and Mantissa

Characteristic is an integral part of logarithm.

**Case 1:**If number is greater than 1.

In this condition, characteristic is considered as one less than the number of digits in the left of decimal point in the given number.

**Example:**256.23

Number of digits to the left of decimal point are 3. Hence, the characteristic is one less than number of digits before decimal points i.e 2.

**Case 2:**If number is less than 1.

In this condition, characteristic is considered as one more than the number of zeros between decimal point and first digit of the number. It is negative and is denoted as

(One bar) 1 or (Tow bar) 2

**Example:**0.00735

Number of zeros between decimal point and first significant digit 7 are 2. Hence the characteristic is one more than number of zeros i.e 3

**Mantissa:**It is the decimal part of logarithm. Log table is used to find the mantissa.

**Quick Tips and Tricks**

**Very Important:**If base is not mentioned, then always remember to take it as 10.

**Properties of logarithm**

1) log

_{x}x = 1

2) log

_{a}1 = 0

3) a

^{logax}= x

4) log_{a}x | 1 |

log_{x}a |

5) log

_{a}(x

^{P}) = p (log

_{a}x)

6) log_{a}x = | log_{b}x | = | log x | - - - (Change of base formula) |

log_{b}a | log a |

7) log_{a} | x | = log_{a}x – log_{b}y - - - - (Division Rule) | ||

y |

8) log

_{a}(xy) = log

_{a}x + log

_{b}y

**Important points to remember**

1) Logarithms are opposite to exponentials which means logs are inverses of exponentials.

2) Given logarithmic form can be converted into exponential form as shown below:

**b = base y = exponent x = answer**

**Exponential form: b**

^{y}= x**Logarithm form: log**

_{b}(x) = y**Log of numbers (2 – 10)**(Must Remember)

Log 2 = 0.301

Log 3 = 0.477= 0.48

Log 4 = 0.60

Log 5 = 0.698 = 0.7

Log 6 = 0.778 = 0.78

Log 7 = 0.845 = 0.85

Log 8 = 0.90

Log 9 = 0.954= 0.96

Log 10 = 1

**Finding log without using calculator**

These are very easy methods to find the log of numbers without calculator.

**Step 1:**You should know all the prime factors of given numbers.

**Step 2: Method 2:**Log of 4 to 9 can be easily determined if only the value of log 2 and log 3 is remembered.

Method 1 | Method 2 |

Log of composite number (x) = Sum of logarithms of its prime factors. Example: Log of 9 =? If you know the value of log 3, the value of log 9 can be easily determined. Log 3 = 0.477 Log of 9 = Sum of logarithms of its prime factors. Log of 9 = log 3 + log 3 = 0.477 + 0.477 = 0.954 Log of 9 = 0.954 | Log 2 = 0.301 Log 3 = 0.477 Log 4 = 2 x log 2 = 0.60206 Log 5 = 1 – log 2 = 0.6989 Log 6 = log 2 + log 3 = 0.778 Log 7 = 0.84510 --- (Remember) Log 8 = 3 x log 2 = 0.9030 Log 9 = 2 x log 3 = 0.9542 |

**Types of Questions**

Type 1: Find the value of given expression.

**Q 1.**log

_{2}64

a. 6

b. 8

c. 16

d. 32

View solution

Correct Option: (a)

We have to simplify the given expression, which means in exponential form. Remember the tricks discussed above. **Hint: ****Logarithm form: log _{b} (x) = y**

**Exponential form: b**

^{y}= x**Given:**log

_{2}64

Let the solution of log

_{2}64 be y.

Log

_{2}64 = y

2

^{y}= 64

2

^{6}= 64

Therefore, y = 6

Q 2. log_{7} | 1 | ||

2401 |

a. 7

b. -3

c. - 4

d. 9

View solution

Correct Option: (c)

We have to simplify the given expression, which means in exponential form. Remember the tricks discussed above. **Hint: ****Logarithm form: log _{b} (x) = y**

**Exponential form: b**

^{y}= xLet the solution of log_{7} | 1 | be y | ||

2401 |

log_{7} | 1 | = y | ||

2401 |

7 ^{y} = | 1 | = | 1 | = 7^{–4} | ||||

2401 | 7^{4} |

Therefore,

y=-4

**Q 3.**49

^{log74}

a. 7

b. 14

c. 16

d. 18

View solution

Correct Option: (c)**Hint: **

a^{logax} = x, here the base of log_{a}x and number which is based to log_{a}x are same. i.e. a

Therefore, in the given expression

49^{log74} = (7^{2})^{log74}

log_{a}(x^{p}) = p(log_{a}x)

Carefully understand the simplification done using the properties of logarithm.

(7 ^{2}) ^{ log74} = (7) ^{(log74)2} = (7) ^{ log7(4 2)}

We know that, a^{logax} = x

Hence, (7) ^{ log7(42)} = 4^{2} = 16