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Characteristics and Mantissa

The integral part of logarithm is called 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 log27 (y) + 2 log 9(81y) = 20, then y is equal to
  1. 1/7
  2. 81
  3. 729
  4. 243
Solution
5 log 27(y) + 2 log 9(81y) = Description: 1242.pnglog3 (y) + Description: 1251.png log3 (81y)
= log 3(81) + logDescription: 1260.png
= 4 +Description: 1269.png log3(y) = 20
 
Hence, log3 y = 6
So, y = 36 = 729
 
Alternatively, this question can be solved by using options too.
 
Example-2
If x ≥ y and y > 1, then the value of the expression Description: 1278.png can never be
  1. –1
  2. –0.5
  3. 0
  4. 1
Solution
P =Description: 1287.png
= logx X − logxy + logyy – logyy
= 2 – logxy − logyx
 
Assume now that logxy = t
Description: 1291.png
 
This can never be equal to 1.
 




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