# Logarithms

 From: Nadeen Dahl Date: 16 March 1999 Subject: What is the difference between log and ln ? My calulator has got two buttons log and ln. What is the difference? Our teacher calls them both "log" or "logarithm". Are they the same thing?

### Maths Help suggests:

No, they are not the same. They are both logarithms, but different kinds of logarithm.

Strictly speaking, a logarithm should have a specified base. This is usually denoted by a small number subscript. For example:
log10 means "log base 10"
log2 means "log base 2"
and so on.

The reason for this is that a logarithm to a certain base is the inverse function of that base raised to a power. (I know that sounds rather complicated...). Put another way: the logarithm, to a given base, of a number is equal to the power to which that base has to be raised to give that number. Consider the following examples:

 log10 1000 = 3 means that 1000 = 103 log2 32 = 5 means that 32 = 25 log3 9 = 2 means that 9 = 32 log4 8 = 1.5 means that 8 = 41.5

In general, if you have the statement
logb A = N
then you can deduce
A = bN
and vice-versa.

So, having revised the general theory of logarithms, let's answer your question.....

The button called log on your calculator is actually for logarithms base 10. Perhaps it should have been labelled log10. You can check this by using it to evaluate log 100. You know the answer is going to be 2 because 100 = 102.

The button called ln is often known as the "natural logarithm", and is actually for logarithms base e. Remember that e is the special number 2.71828....

So which one should you use?

If the question contains the exponential base e then use the natural logarithm ln when rearranging or solving the equation.

For example:
 Find the value of t for which 50e - 0.4t = 10 First divide by 50 e - 0.4t = 0.2 Take natural logs of both sides -0.4t = ln0.2 = -1.6094 Divide by -0.4 t = 4.02 (2 d.p.)

(For equations which do not include the exponential function e, then you can use the other log button if you prefer.)

The other situation where it is essential to use ln is for integration and differentiation. The standard result:

uses the natural logarithm.