Introduction to Logarithms
In its simplest form, a logarithm answers the question:
How many of one number do we multiply to get another number?
Example: How many 2s do we multiply to get 8?
Answer: 2 × 2 × 2 = 8, so we had to multiply 3 of the 2s to get 8
So the logarithm is 3
How to Write it
We write "the number of 2s we need to multiply to get 8 is 3" as:
log_{2}(8) = 3
So these two things are the same:
The number we multiply is called the "base", so we can say:
 "the logarithm of 8 with base 2 is 3"
 or "log base 2 of 8 is 3"
 or "the base2 log of 8 is 3"
Notice we are dealing with three numbers:
 the base: the number we are multiplying (a "2" in the example above)
 how often to use it in a multiplication (3 times, which is the logarithm)
 The number we want to get (an "8")
More Examples
Example: What is log_{5}(625) ... ?
We are asking "how many 5s need to be multiplied together to get 625?"
5 × 5 × 5 × 5 = 625, so we need 4 of the 5s
Answer: log_{5}(625) = 4
Example: What is log_{2}(64) ... ?
We are asking "how many 2s need to be multiplied together to get 64?"
2 × 2 × 2 × 2 × 2 × 2 = 64, so we need 6 of the 2s
Answer: log_{2}(64) = 6
Exponents
Exponents and Logarithms are related, let's find out how ...
The exponent says how many times to use the number in a multiplication. In this example: 2^{3} = 2 × 2 × 2 = 8 (2 is used 3 times in a multiplication to get 8) 
So a logarithm answers a question like this:
In this way:
The logarithm tells us what the exponent is!
In that example the "base" is 2 and the "exponent" is 3:
So the logarithm answers the question:
What exponent do we need
(for one number to become another number) ?
The general case is:
Example: What is log_{10}(100) ... ?
10^{2} = 100
So an exponent of 2 is needed to make 10 into 100, and:
log_{10}(100) = 2
Example: What is log_{3}(81) ... ?
3^{4} = 81
So an exponent of 4 is needed to make 3 into 81, and:
log_{3}(81) = 4
Common Logarithms: Base 10
Sometimes a logarithm is written without a base, like this:
log(100)
This usually means that the base is really 10.
It is called a "common logarithm". Engineers love to use it.
On a calculator it is the "log" button.
It is how many times we need to use 10 in a multiplication, to get our desired number.
Example: log(1000) = log_{10}(1000) = 3
Natural Logarithms: Base "e"
Another base that is often used is e (Euler's Number) which is about 2.71828.
This is called a "natural logarithm". Mathematicians use this one a lot.
On a calculator it is the "ln" button.
It is how many times we need to use "e" in a multiplication, to get our desired number.
Example: ln(7.389) = log_{e}(7.389) ≈ 2
Because 2.71828^{2} ≈ 7.389
But Sometimes There Is Confusion ... !
Mathematicians use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:
Example  Engineer Thinks 
Mathematician Thinks 


log(50)  log_{10}(50)  log_{e}(50)  confusion 
ln(50)  log_{e}(50)  log_{e}(50)  no confusion 
log_{10}(50)  log_{10}(50)  log_{10}(50)  no confusion 
So, be careful when you read "log" that you know what base they mean!
Logarithms Can Have Decimals
All of our examples have used whole number logarithms (like 2 or 3), but logarithms can have decimal values like 2.5, or 6.081, etc.
Example: what is log_{10}(26) ... ?
Get your calculator, type in 26 and press log Answer is: 1.41497... 
The logarithm is saying that 10^{1.41497...} = 26
(10 with an exponent of 1.41497... equals 26)
This is what it looks like on a graph: See how nice and smooth the line is. 
Read Logarithms Can Have Decimals to find out more.
Negative Logarithms
−  Negative? But logarithms deal with multiplying. What is the opposite of multiplying? Dividing! 
A negative logarithm means how many times to divide by the number.
We can have just one divide:
Example: What is log_{8}(0.125) ... ?
Well, 1 ÷ 8 = 0.125,
So log_{8}(0.125) = −1
Or many divides:
Example: What is log_{5}(0.008) ... ?
1 ÷ 5 ÷ 5 ÷ 5 = 5^{−3},
So log_{5}(0.008) = −3
It All Makes Sense
Multiplying and Dividing are all part of the same simple pattern.
Let us look at some Base10 logarithms as an example:
Number  How Many 10s  Base10 Logarithm  

.. etc..  
1000  1 × 10 × 10 × 10  log_{10}(1000)  = 3  
100  1 × 10 × 10  log_{10}(100)  = 2  
10  1 × 10  log_{10}(10)  = 1  
1  1  log_{10}(1)  = 0  
0.1  1 ÷ 10  log_{10}(0.1)  = −1  
0.01  1 ÷ 10 ÷ 10  log_{10}(0.01)  = −2  
0.001  1 ÷ 10 ÷ 10 ÷ 10  log_{10}(0.001)  = −3  
.. etc.. 
Looking at that table, see how positive, zero or negative logarithms are really part of the same (fairly simple) pattern.
The Word
"Logarithm" is a word made up by Scottish mathematician John Napier (15501617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "rationumber" !