Introduction to Logarithms

In its simplest form, a logarithm answers the question:

How many of one number multiply together to make another number?

Example: How many 2s multiply together to make 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 it like this:

log2(8) = 3

 So these two things are the same:

logarithm concept 2x2x2=8 same as log_2(8)=3

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 base-2 log of 8 is 3"

Notice we are dealing with three numbers:

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

Exponents and Logarithms are related, let's find out how ...

2 cubed

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

In this example: 23 = 2 × 2 × 2 = 8

(2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

2 with what exponent = 8

In this way:

2^3=8 becomes log_2(8)=3

The logarithm tells us what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:

2^3=8 becomes log_2(8)=3

So the logarithm answers the question:

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

The general case is:

a^x=y becomes log_a(y)=x 

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

102 = 100

So an exponent of 2 is needed to make 10 into 100, and:

log10(100) = 2

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

34 = 81

So an exponent of 4 is needed to make 3 into 81, and:

log3(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.

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 we need to use 10 in a multiplication, to get our desired number.

Example: log(1000) = log10(1000) = 3

 

Natural Logarithms: Base "e"

Another base that is often used is e (Euler's Number) which is about 2.71828.

calculator ln button

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) = loge(7.389) ≈ 2

Because 2.718282 ≈ 7.389


But Sometimes There Is Confusion ... !

Mathematicians may 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!

 

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 log10(26) ... ?

log

Get your calculator, type in 26 and press log

Answer is: 1.41497...

The logarithm is saying that 101.41497... = 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.

  log 10 of 26

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 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
10 times 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..      

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 (1550-1617), from the Greek word logos meaning "proportion, ratio or word" and arithmos meaning "number", ... which together makes "ratio-number" !

340, 341, 2384, 2385, 2386, 2387, 3180, 3181, 2388, 2389