Important Concepts and Formulas - Logarithms

Important Concepts and Formulas - Logarithms

• Logarithm Basics   

if y=logbx , then by=x

where logbx=y = log to the base b of x
Please note that b(base) is a positive positive real number, other than 1.
if x=by , then logbx=y

where logbx=y = log to the b b of x
Please note that b(base) is a positive positive real number, other than 1.
Example
16=24 (in this expression, 4 is the power or the exponent or the index and 2 is the base)
Hence we can say that log216=4 (i.e., log to the base 2 of 16 = 4)
In other words, both 16=24 and log216=4 are equivalent expressions.
If base = 10, then we can write logx instead of log10x
logx is called as the common logarithm of x

If base = e, then we can write lnx instead of logex
lnx is called as the natural logarithm of x


Please note that e is a mathematical constant which is the base of the natural logarithm. It is known as Euler's number. It is also called as Napier's constant

e=1+11+11.2+11.2.3+11.2.3.4+⋯≈2.71828
ex=1+x+x22!+x33!+x44!+⋯



• Logarithms - Important Properties   

• logb1=0        (∵ b0=1)



• logbb=1        (∵ b1=b)



• y=lnx⇒ey=x



• x=ey⇒lnx=y



• x=lnex=elnx



• blogbx=x



• logbby=y



• Laws of Logarithms   

• logbMN=logbM+logbN (where b, M, N are positive real numbers and b ≠ 1)



• logbMN=logbM−logbN(where b, M, N are positive real numbers and b ≠ 1)



• logbMc=c logbM (where b and M are positive real numbers , b ≠ 1, c is any real number)



• logbM=logMlogb=lnMlnb=logkMlogkb (where b, k and M are positive real numbers, b ≠ 1, k ≠ 1)



• logba=1logab (where a and are positive real numbers, a ≠ 1, b ≠ 1)



• Let logbM=logbN. Then M=N where b, M and N are positive real numbers and b ≠ 1.

• Mantissa and Characteristic   

The logarithm of a number has two parts, known as characteristic and mantissa.

1. Characteristic
The internal part of the logarithm of a number is called its characteristic.

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.

Examples
Number	Characteristic
612.25	2
16.291	1
2.1854	0
0.9413	1¯
0.03754	2¯
0.00235	3¯

2. Mantissa
The decimal part of the logarithm of a number is known is its mantissa. We normally find mantissa from the log table.