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:
 |
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:
- 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 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 discover how ...
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 an exponent gives you this answer:
And a logarithm gives you this answer:
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 log10(100) ... ?
102 = 100
So an exponent of 2 is needed to make 10 into 100:
log10(100) = 2
Example: What is log3(81) ... ?
34 = 81
So an exponent of 4 is needed to make 3 into 81:
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.
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...
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, and so on.
Example: what is log10(26) ... ?
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 plot:
See how nice and smooth the line is.
Read Logarithms Can Have Decimals to discover 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.
Example: What is log8(0.125) ... ?
Well, 1 ÷ 8 = 0.125,
So log8(0.125) = −1
We may need 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 | ||
|---|---|---|---|---|
 |
.. and so on.. | |||
| 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 | |
| .. and so on.. | ||||
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 Middle Latin "logarithmus" meaning "ratio-number" !