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Negative characteristics

  • All numbers lying between 1 and 0.1 have logarithms lying between 0 and -1. Since decimal part is always written positive, the characteristic is -1.
  • All numbers lying between 0.1 and 0.01 have their logarithms lying between -1 and -2 as the characteristic of their logarithms.

Note: In general, the logarithm of a number having n zeros just after the decimal point is (n + 1) + a fraction.


Find the characteristic and mantissa in log(0.025) = -1.602.
Since the value of logarithm is negative, we cannot directly write -1 as the characteristic and -0.602 as the mantissa.

We will try to convert the decimal part into a positive quantity. Adding and subtracting 2 to -1.602, we get


log(0.025) = -1.602 + 2 - 2 = (2 - 1.602) - 2 = -2 + 0.398

Now we have the decimal part as positive, so the characteristic becomes -2 and mantissa is 0.398. It is also written as Description: 13358.png+ 0.398 (read as 2 bar + 0.398).


Note: Remember that logarithms and indices are transformation of one another.

  • In indices, we learnt that any number raised to 0 is 1,
    i.e., a0 = 1 Description: 13479.png
    i.e., logarithm of 1 to any base is zero
  • We also know that any number raised to unity is itself,
    i.e., a1 = a Description: 13488.png
    i.e., logarithm of any number to the same base is unity

Find the value of Description: 13500.png
Converting the given expression into division of two logarithms with the same base, we can write
Description: 13542.png 
If log2 (log2 x= 2, then x =?
We can use the definition of logarithms and simplify by converting the given expression into exponential form.
Description: 13553.png 


If Description: 13559.png, then find Description: 13565.png
If loga b = c, therefore Description: 13693.png


The value of Description: 13703.png
We have multiple log terms with different bases. Let’s use change of base to write all terms with a common base.
Description: 13709.png 
Description: 13715.png


Find the value of log10 5, where log10 2 = 0.3010.
Description: 13746.png
Description: 13758.png 

Description: 13764.png
It will be easier to solve such problems if the bases are all equal.
Applying the change of base rule to all the terms, we get,
Description: 13783.png 
Description: 13789.png

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