LOGARITHMS
Logarithm : John Napier a great mathematician in 1614 invented logarithm.
These provide a very fast method of mathematical calculations.
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Logarithm is defined as ax = m then log m = x or log m = x thenax = m
a
First is called the log form and the second is called exponetial or index form.
Some Important Conclusions :
- x0 = 1 for all x > 0 then log 1 = 0 Thus log of 1 to any base is zero.
- x = x then logx x = Thus log for any number to the same base in one.
LAWS OF LOGARITHMS
- Product Rule : logn y = lognx + logn y
x
- Quotient Rule : logn — = logn x – logn y
y
- Power Rule : logn x = y log x
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y
4. Change of Base Rule :
logn x logn x = ——–
logn y
logn x. logm y. log1 z = log1 x logy x. logx y = 1
[logx x = 1]
1
or logy x can be written as—— –.
logx y
- Logarithm of a root of a number is the logarithm of the number divided by the
index of the root.
logn mÖx = log
l
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xl/m = — log x. m
- The logarithm of 1 to any base is zero e. loga 1 = 0.
- The logarithm of any quantity to the same base is unity, e., logx x = 1.
- Logarithms to the base 10 are called common
- If no base is indicated the base is understood to be 10. Hence log 10 = 1, log 100 = 2, log 1000 = 3,
LOGARITHMS WITH FOUR FIGURE TABLES ONLY
Common Logarithms : Logarithms calculated to the base 10 are known as common logarithms. These consists of two parts :
- Characteristic [the integral or whole number part]
- Mantissa [the fractinal or decimal part]
Characteristic : 1. To find the characteristic of a number greater than one. “Characteristic is one less than the number of digits to the left of the decimal point in the given number.” e.g. characterstic of 514 is 2 and 3125 is 3.
- To find the characteristic of a number less than one. “Characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and is negative.
Mantissa : Mantissa of a number is found with the help of logarithmic tables.
- The mantissa is the same for the logarithms of all numbers having the same significant
- The logarithm of one digit number, say 2, is to be seen in the table, opposite
to 20.
- The mantissa is always taken
Antilogarithm : If log a = m, then a = antilog of m. i.e., The number correspond-
ing to a given logarithm is called antilogarithm.
- If the characteristic of the logarithm is positive, then : “put the decimal point after (n + 1)th digit, where n is equal to ”
- If the characteristic of the logarithm is negative, then : “put the decimal point so that the first significant digit is at ‘n’th place, where n = ”
Some Important Points : Those logarithms whose base is 10 are known as decimal logarithms while which has base e(e = 2.71828…) are known as natural or Napierian logarithms. Natural logarithm is changed to decimal logarithm as
log b = Ln b loge
logb
Ln b =——- [loge = 0.434294]
loge
Uses of Logarithms :
- In calculating compound
- In calculating population growth
- In calculating depreciation
- In calculations of
