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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

log2(8) = 3

So these two things are the same:
logarithm concept

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 base-2 log of 8 is 3"

More Examples

Example: What is log5(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: log5(625) = 4


Example: What is log2(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: log2(64) = 6

Exponents

Logarithms are exponents!

8 to the Power 2

The exponent of a number says how many times to use the number in a multiplication.

In this example: 82 = 8 × 8 = 64

  • In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

 


So this: logarithm concept
is also this: logarithm concept

So a logarithm also answers the question

What exponent do we need
(for one number to become another number)
?


Example: What is log10(100) ... ?

102 = 100, so an exponent of 2 is needed to make 10 into 100


Answer: log10(100) = 2

Example: What is log3(81) ... ?

34 = 81, so an exponent of 4 is needed to make 3 into 81


Answer: log3(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.

log

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) = log10(1000) = 3

Natural Logarithms: Base "e"

Another base that is often used is e (eulers number) which is approximately 2.71828.

ln

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) = loge(7.389) ≈ 2

Because 2.718282 ≈ 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) log10(50) loge(50) confusion
ln(50) loge(50) loge(50) no confusion
log10(50) log10(50) log10(50) no confusion

So, be careful when you read "log" that you know what base they mean!

Negative Logarithms

negative 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 log8(0.125) ... ?

Well, 1 ÷ 8 = 0.125, so log8(0.125) = -1

Or many divides:

Example: What is log5(0.008) ... ?

1 ÷ 5 ÷ 5 ÷ 5 = 5-3, so log5(0.008) = -3

It All Makes Sense

Multiplying and Dividing are all part of the same simple pattern.

Let us look at some Base-10 logarithms as an example:

  Number How Many 10s Base-10 Logarithm
larger-smaller .. etc..        
1000 1 × 10 × 10 × 10 log10(1000) = 3
100 1 × 10 × 10 log10(100) = 2
10 1 × 10 log10(10) = 1
1 1 log10(1) = 0
0.1 1 ÷ 10 log10(0.1) = -1
0.01 1 ÷ 10 ÷ 10 log10(0.01) = -2
0.001 1 ÷ 10 ÷ 10 ÷ 10 log10(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.