# 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.

Example
Find the characteristic and mantissa in log(0.025) = -1.602.
Solution
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 + 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

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

i.e., logarithm of any number to the same base is unity

Example
Find the value of
Solution
Converting the given expression into division of two logarithms with the same base, we can write

Example
If log2 (log2 x= 2, then x =?
Solution
We can use the definition of logarithms and simplify by converting the given expression into exponential form.

Example
If , then find
Solution
If loga b = c, therefore

Example
The value of
Solution
We have multiple log terms with different bases. Letâ€™s use change of base to write all terms with a common base.

Example
Find the value of log10 5, where log10 2 = 0.3010.
Solution

Example
Solution
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,