Basic Properties of Logarithm

Chinmay Pisu

Basic Properties of Logarithm

This article is basically for all those students who have just entered in 10^th/11^th classes. When you start working on some difficult level questions on Mathematics, use of “Logarithm” comes automatically. However, only a basic knowledge of Logarithm is required to gear up initially which I am providing as below :

Definition of Log : log_y x = z means x = y^z

log_y x is pronounced as “log of ‘x’ to the base ‘y’ “

Note : a. ’x’ and ‘y’ MUST be positive.

b. In most of the cases, “base” (=y) is either “10″ or “e” (Numerically, e=2.718).

Basic Properties of Log :
1. log_y 1 = 0 (where ‘y’ is any base value)
2. log_y y = 1 (where ‘y’ is any value) => log₁₀ 10 = 1
3. log_y (a * b) = log_y a + log_y b
4. log_y (a / b) = log_y a - log_y b
5. log_y (x)^m = m log_y x => log_y (1/x) = - log_y x
6. log_e x = 2.303 log₁₀ x
  Note : Many a times, log_e x is written as ln x (ln = Natural Log)
7. log_a x < log_a y where 0 < x < y
  
  Note : The above means that the graph of log is a strictly increasing function.
Logarithmic series :

Logarithm of any number can be represented in terms of a Mathematical expression (known as Series).

ln (1 + x) = x – x²/2 + x³/3 – + …… (infinite terms) [Learn this series]

Note : The above series is a convergent series which means that for a given finite ‘x’, the series will provide a finite value for ln (1 + x).

In cases where “x” (above) is very small compared to “1″, you can also write as :

ln (1 + x) = x
Values to be learned :

Even upto 12^th class level, following values of log₁₀ are necessary and sufficient :

log₁₀ 1 = 0        log₁₀ 2 = 0.301        log₁₀ 3 = 0.477        log₁₀ 4 = 0.602

log₁₀ 5 = 0.699        log₁₀ 6 = 0.778        log₁₀ 7 = 0.845        log₁₀ 8 = 0.903

log₁₀ 9 = 0.954        log_e 2 = 0.693

Try yourself to calculate log₁₀ values for the followings :

log₁₀ 15,    log₁₀ 1.004,    log₁₀ 25,    log₁₀ 100,    log₁₀ (1/30),     log₁₀ (Sqrt (2))