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
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:
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!
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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"
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| So this: |
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| is also this: |
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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.
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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.
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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 |
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| 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
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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
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