Logarithms

Index laws revision

There are three main laws you can use when you are working with indices.  ‘a’, ‘b’, ‘m’, and ‘n’ are numbers.  They don’t have to be integers, they can be decimals or fractions, and can be negative.

1.  a^m ´ aⁿ = a^m+n

                                                          

2.  (a^m)ⁿ = a^mn

                                                            

3.  (ab)^m = a^mb^m

                                                      

Remember that anything to the power 0 is 1.

Logarithms are another way of writing indices and powers.  Take for instance the following statement:

                                                           

This can be rewritten as a log statement as follows:

                                                        

This log statement says exactly the same thing, that 8 to the power 2 equals 64.  In general, if we have:

                                                         

This statement says that the base a to the power c equals b:

                                                            

If I had a question saying:

                                                        

I would have to work out what power I have to raise 2 by to get 32. The answer is 5.  So:

                                                        

There are two types of logarithms that have a special base.

Common logarithms

These all have a base of 10.

These are the easiest to work with in general.  For instance:

                                                        

                                                       

                                                      

Natural logarithms

These all have the special number ‘e’ as a base, which is equal to about 2.718. It is a number that is very important in calculus. Natural logarithms are also expressed using ln instead of log.  For example: