# Characteristics and Mantissa

The integral part of logarithm is called

â€‹

*Characteristic*and its decimal part is called*Mantissa*. Logarithms to the base 10 are called*Common logarithms*. The characteristic of common logarithm can be found out by a visual inspection. The characteristics of the logarithm (base 10) of a number greater than 1 is less by one than the number of digits in the integral part and is a positive. However, if a decimal fraction number is less than 1 but positive, its characteristic will be greater by unity than the number of consecutives zeros immediately after the decimal point and is a negative.Example-1

If 5 log

_{27}(*y*) + 2 log_{9}(81*y*) = 20, then*y*is equal to- 1/7
- 81
- 729
- 243

Solution

5 log

= log

= 4 + log
Hence, log

So,
Alternatively, this question can be solved by using options too.

_{27}(*y*) + 2 log_{9}(81*y*) = log_{3}(*y*) + log_{3 }(81*y*)= log

_{3}(81) + log= 4 + log

_{3}(*y*) = 20_{3 }*y*= 6So,

*y*= 3^{6}= 729Example-2

If

*x*â‰¥*y*and*y*> 1, then the value of the expression can never be- â€“1
- â€“0.5
- 0
- 1

Solution

P =

= log

= 2 â€“ log
Assume now that log

This can never be equal to 1.

= log

_{x}*X*âˆ’ log*+ log*_{x}y*â€“ log*_{y}y_{y}y= 2 â€“ log

*âˆ’ log*_{x}y_{y}x*=*_{x}y*t*