Logarithms
In its simplest form, a logarithm answers a simple question:
How many of one number do we multiply to get another number?
Example
How many 2s need to be multiplied to get 8?
Answer: 2 × 2 × 2 = 8, so we needed to multiply 3 of the 2s to get 8
So the logarithm is 3
Notice we are dealing with three numbers:
 the number we are multiplying (a "2")
 how many times to use it in a multiplication (3 times)
 The number we want to get (an "8")
How to Write it
We would write "the number of 2s you need to multiply to get 8 is 3" as
log_{2}(8) = 3
So these two things are the same:
The "Base"
The number we are multiplying is called the "base", so we would 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"
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
Logarithms are exponents!

The exponent of a number says how many times to use the number in a multiplication.
In this example: 8^{2} = 8 × 8 = 64
 In words: 8^{2} could be called "8 to the second power", "8 to the power 2" or
simply "8 squared"

So this: 

is also this: 

So a logarithm also answers the question
What exponent do we need
(for one number to become another number) ?
Example: What is log_{10}(100) ... ?
10^{2} = 100, so an exponent of 2 is needed to make 10 into 100
Answer: 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
Answer: log_{3}(81) = 4
Common Logarithms: Base 10
Sometimes you will see a logarithm 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 you need to use 10 in a multiplication, to get the desired number.
Example: log(1000) = log_{10}(1000) = 3
Natural Logarithms: Base "e"
Another base that is often used is e (eulers number) which is approximately 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 you need to use "e" in a multiplication, to get the 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!
Negative Logarithms

Negative? But logarithms deal with multiplying. What could be the opposite of multiplying? Dividing! 
A negative logarithm means how many times to
divide by the number.
We could 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.. 




If you look at that table, you will see that positive, zero or negative logarithms are really part of the same (fairly simple) pattern.
