Logrithms

Logarithm: If a is a positive real number, other than 1 and aⁿ = X, then we write:

n = log(a) X and we say that the value of log X to the base a is n.

Example:1. 10³ = 1000 => log₁₀1000 = 32. 2ˉ̄³ = 1/8 => log₂ 1/8 = - 33. 3⁴ = 81 => log₃ 81=44. (.1)² = .01 => log·₁ (.01) = 2.

IMPORTANT FACTS

The logarithm of a number contains two parts, namely characteristic and mantissa.

Characteristic: The integral part of the logarithm of a number is called its characteristic.

Mantissa: The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.

Characteristic & Mantisaa

Case I: When the number is greater than 1. In this case, the characteristic is one less than the

number of digits in the left of the decimal point in the given number.

Case II: When the number is less than 1. In this case, the characteristic is one more than the

number of zeros between the decimal point and the first significant digit of the number and it is negative.

Instead of - 1, - 2, etc. we write, 1 (one bar), 2 (two bar), etc.

Two cases of Characteristic

Example:Number Characteristic Number

Characteristic348.25 2 0.6173 `1 (one

bar)46.583 1 0.03125 `2 (two

bar)9.2193 0 0.00125 `3 (three

bar)

1. log(a)(XY) = log(a)X + log(a)Y2. log(a) (X/Y) = log(a)X – log(a)Y 3. log(x) X=14. log(a) 1 = 05. log(a)(Xⁿ)=n log(a)(X) 6. log(a)X =1/log(x)A7. log(a)X = log(b) X/log(b)A=log X/log A

Remember: When base is not mentioned, it is taken as 10.

Common Logarithms: Logarithms to the base 10 are known as common

logarithms.

Properties of Logarithms:

Evaluate: 1. log₃27 2. log₇(1/343) 3. log₁₀₀(0.01) 4. log₅3*log₂₇25 5. log ₉27 –log₂₇9 If log 2 = 0.3010 and, the value of log5 512

is