
A Recap of Laws of Indices
Consider the following expression:
A shorter version of this is . We say that 2 raised to 5 is . By this, we mean multiplying the number by itself times. The number is termed as the index.
Whenever the index is a negative integer, i.e. , we actually mean
Whenever the index is a reciprocal of any integer, i.e. , we actually mean
I) For any number except ,
II) For any number ,
III) For any number ,
IV) For any number ,
V) For any number ,

Logarithms
Logarithms are just another way of representing a number raised to an index. Let and , then
This is known as the logarithmic form and the first representation is known as the indicial form.
is known as the logarithm (short form log) and is the base.

Important Points
I) Logarithms of negative numbers and zero are NOT defined.
II) Logarithm of to any valid base is always .
III) Logarithm of any number to the same (valid) base is .
IV) This property is interesting:
It proves that the logarithmic function and the exponential function are inverses of each other.
V) If and , then
VI) If , then .

Laws of Logarithms
If are positive real numbers and , then
Let be any real number such that , then

Logs and Antilogs
We know that gives . We write this alternatively as equal to antilog of to the base .

2 Important Bases, and
Commonly used bases of logarithms are and . The logarithms to base are generally written as log and those to base are generally written as ln.
Note: is an irrational number, its approximate value is .

Reading a Logarithmic Table
Standard log tables are available, with base . The log tables are handy, when it comes to multiplication, division, powers, roots of numbers. However, today, they’ve almost been replaced by electronic calculators.
Log of a number: The log contains 2 parts, the characteristic, the integral part and the mantissa, the decimal part.
I) The characteristic of any number is the power of , closest to the number, and smaller than that number. e.g., characteristic of will be , because is the power of , which is closest to and is smaller than .
can be written as . Hence, the characteristic is . Similarly, characteristic of will be ; . Characteristic of will be ; it can be written as .
Thus, if a number is written in a form, such that there is a single digit nonzero number to the left of decimal point, the power of will be its characteristic.
II) The mantissa of a number is to be found out from log table. Irrespective of the position of decimal point, we look for the 1st nonzero digit of the number. The first 2 digits form a number. We look for a row, containing this number. The column number is the 3rd digit. The 4 digit number sharing the row and the column is the mantissa.
If the number contains more than 3 digits, the mean differences are subtracted/added as per the instructions given in the log table. Note that \textbf {mantissa is always positive}.
III) The logarithm is characteristic + mantissa.
IV) The last step is taking antilogs. We make sure that the decimal part of the number so obtained is positive. Now, we look in antilog table and search for the rows and columns corresponding to the mantissa. Procedure is similar. (1st 2 digits – Row, 3rd digit – Column, 4th digit – Mean Differences)
V) The decimal point is put at a place, such that the final number has characteristic equal to the integral part of the logarithm.
Example:
Characteristic and mantissa .
Hence,
is closer to . So, logarithms give us approximate answers. The error is , which is acceptable.
For , we will divide the log of by and then take its antilog, as .