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Introduction to Logarithms
In its simplest form, a logarithm answers the question:
Introduction
Introduction to
to Logarithms
Logarithms
How many of one number do we multiply to get another number?
Example: How many 2 s do we multiply to get 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 "the number of 2s we need to multiply to get 8 is 3" as:
log2(8) = 3
So these two things are the same:
2x2x2 =8 log (8 ) = 3
2
3
base
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 find out how ...
exponent
The exponent says how many times to use the number in a multiplication.
3
2 In this example: 23 = 2 × 2 × 2 = 8
(2 is used 3 times in a multiplication to get 8)
base
So a logarithm answers a question like this:
In this way:
The logarithm tells us what the exponent is!
In that example the "base" is 2 and the "exponent" is 3:
exponent
23 = 8 log2( 8) = 3
base
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, 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.
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 use "log" (instead of "ln") to mean the natural logarithm. This can lead to confusion:
Engineer Mathematician
Example
Thinks 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) ... ?
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.
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
.. 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" !
Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7
Question 8 Question 9 Question 10
Exponents, Roots and Logarithms
Working with Exponents and Logarithms
Algebra Index
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