Ifthenand we say we are 'finding log baseof'.
For example ifthento 3 decimal places.
Some bases are more commonly used than others. Base 10 is the most commonly used. This is because we have 10 fingers, so the most commonly used number system is in base 10. Some cultures use other bases. There is a tribe in West Africa that uses base 20 – 20 fingers and toes – and our time system uses base 60: 60 minutes in a hour, 60 seconds in a minute.
In maths though, the most natural base is
This number arises naturally because the solution to the differential equationis
In some senses the base is not really relevant since an exponential can be expressed in any base. Ifthen we can writeand the base is then 2.
Bases less than one are also possible. Ifthen the base isWe can writeandso
All bases have in common though that they must be positive, reflecting that the equationhas no solution in general if
Exponential functions with different bases are shown in the graph below.
]]>If however, the interest is compounded more regularly, then something a little bit strange happens. Suppose £1000 is invested at 12% per annum. If interested is compounded annually then at the end of a year, the original £1000 will have grown to £1120. If however, it is compounded monthly, then the monthly rate of interest will be 12/12 =1% and after 1 year the original £1000 will have grown to
In fact if the year is divided intotime periods, so that interest is compounded n times a year, the interest per time period isand the amount of money will have grown to
The table below shows the investment after 1 year for various values of n.
10 
1126.691779 
100 
1127.415743 
1000 
1127.488731 
10000 
1127.495196 
100000 
1127.495975 
Astends to infinity, this expression tends to a limit
We can generalise this reasoning, so that if annual interest ofis compounded continuously on an investment ofat the end of a year the investment will have grown to and at the end ofyears the principal will have grown to
]]>The relationship between powers of 10 and logarithms in base 10 is illustrated in the following table.
Number 
Written as a Power of 10 

10,000 
4 

1,000 
3 

100 
2 

10 
1 

0.1 
1 

0.01 
2 
This can be extended in a natural way to roots and non – integer powers of 10.
In fact all positive numbers can be written as a power of 10 using the relationship and as a logarithm using the relationshipand any number written in standard form asreturn the logarithm
]]>is irrational and the decimal expansion ofcontinuous forever with no pattern, although well known methods exist for calculatingto however many decimal places are desired.
arises naturally in maths, when the rate of change of something is proportioal to the quantity present.
Suppose the rate of change of a population
If the rate of change ofis proportional towe can writeThis is a differential equation and can be integrated to givewhereis the initial population andis the number given above. . In particular ifthe rate of change of the population is equal to the population and the population will grow by a face ofin unit time period. Whatever the value ofas long asthe growth ofis exponential meaningincreases by a constant factor in each time period (and if the value ofdecreases by a constant factor in each time period).
Logarithms with base e obey the same log rules as all other logs, but the number e is special enough for any log with base e to have a special name. They are called natural logs – or logarithme naturel from French and ln for short  so that
The number e appears in every branch of maths, from number theory, complex numbers, trigonometry, differential equationsm,,, and is one of the most important constants in maths, alongside the numberin significance.
]]>1, 3, 9, 27, 81, 243
The terms of the sequence alternate between positive and negative numbers.
The base is the above expressions is 3.
To solve the equationwithwe can log both sides to obtain
(1) (log here means log base 10 )
This is not a general result for real numbers. It can only be used forsince we cannot take the log of a negative number (at least when keeping to real numbers).
If we try and solve it for the equationapplying (1) we obtainwhich is not defined, since the log of a negative number is not defined (keeping to real numbers). In fact, inspection of the equationreturns the solution
We can however solve the equation by taking log base (3) of both sides. If we do this for the equationwe obtain
Now use the identityifto giveto give
This method works in this particular case, but not generally.
The equationhas no solution in real numbers with any amount of manipulation. Solutions involving complex numbers do exist however.
]]>(1)
(2)
(3)
(4)
(5)
Often we can simplify a logarithmic expression so that it becomes either a single log, or a constant plus a log.
Example: Simplify
using (1)
using (3)
using (5)
Example: Ifexpressterms of p
using (1)
using (3)
Simplify
using (2)
using (5)
Simplify
from (2)
]]>