# 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 (

**+ 1) +**

*n***fraction.**

*a*

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.
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).

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

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

- In indices, we learnt that any number raised to 0 is 1,
^{0}= 1 - We also know that any number raised to unity is itself,
^{1}= a

Example

Find the value of

Solution

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

Example

If log

_{2}(log_{2}*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 log

_{a}*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 log

_{10}5, where log

_{10}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,