Basic Properties of Logarithm
This article is basically for all those students who have just entered in 10th/11th 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 : logy x = z means x = yz
logy 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 :
- logy 1 = 0 (where ‘y’ is any base value)
- logy y = 1 (where ‘y’ is any value) => log10 10 = 1
- logy (a * b) = logy a + logy b
- logy (a / b) = logy a - logy b
- logy (x)m = m logy x => logy (1/x) = - logy x
- loge x = 2.303 log10 x
Note : Many a times, loge x is written as ln x (ln = Natural Log)
loga x < loga 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 12th 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 loge 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))