Aptitude Logarithm Concepts and Formulas

General rule: The logarithm of any positive real number y, other than 1 when a^x = y, then component x is called the logarithm of y to the base a, then x= log_a y.

Important terms:

a) The logarithm of one (1) to any base is zero.

Log_a 1 = 0

b) The logarithm of zero (0) to any base greater than unity is -∞.

Log_a  0 = -∞

c) The logarithm of any number (a) to the same base is always unity.

Log_a  a = 1

d) Let log_b x = p, then x= b^p

Or, x = b ^log_b^x

e) The logarithm of product:

Log_a  (m*n) = log_a  m * log_a  n

f) The logarithm of the fraction:

log_a (m/n)= log_a  m - log_a  n

g) Power formula:

Log_a  mⁿ = n log_a  m.

h) Log_b a = log_c a / log_c b

i) Log_b c = 1 / log_c b

j) Log_b a = log_c a * log_b c

k) Log_xⁿ (y^m) = mlog_a y/ nlog_a x

Common logarithm: Logarithms to the base 10 are known as a common logarithms.

Therefore, Log10 10 = 1 is known as a common logarithm.

Characteristic and mantissa:

Every logarithm has two parts: the whole of the integer part and the fraction or decimal parts. The integer part is called the characteristics, and decimal part is called the mantissa.

There are two rules for characteristics:

i. To find the characteristics of a number greater than one.

Characteristic is 1 less than the number of digits to the left of the decimal point in the given numbers.

ii. To find the characteristic of a number less than one.

Characteristic is 1 more than the number of zeros between the decimal point and the 1^st significant digit of the number. The number is represented by a bar because of negative.

For example:

Number	Characteristic	Number	characteristic
6.125	0 (only one digit (6) before the decimal point that?s why characteristic =1-1 =0)	0.6125	͞1 (there is no zero between decimal point, and the first significant digit (here first significant bit is 6). So the characteristic will be 0+1 = 1)
65.23	2-1 = 1	0.06125	1+1 = ͞2
652.3	3-1 = 2	0.006125	2+1 = ͞3
6523	4-1 = 3	0.0006125	3+1 = ͞4