The Logarithms Tutorial - {sectionno}


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Summary

This tutorial describes logarithms: what they are and how they may be used.

Objectives

If you complete this tutorial you should be able to


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1. Raising to a power

You know that if you multiply a number by itself, this is called 'squaring' the number. Thus 16 is 4 x 4, or '4 squared'. If you multiply three copies together you get the number 'cubed'; for example 64 is 4 x 4 x 4, or '4 cubed'. After three copies, there isn't really any terminology, so we use the phrase 'raised to the power of' instead. For example 256 is 4 x 4 x 4 x 4 or '4 raised to the power 4'. Squaring is then just 'raising to the power 2', and cubing is 'raising to the power 3'.

This table show the pattern for the first few powers of some number which we refer to as 'N'

N x N
N raised to the power 2 (N 'squared')
N x N x N
N raised to the power 3 (N 'cubed')
N x N x N x N
N raised to the power 4
N x N x N x N X N
N raised to the power 5

The pattern clearly extends indefinitely to any power.

Given the pattern, answer the following question.

{ASK Number=1}

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2. Power Notation

The use of the phrase 'raised to the power of' is rather unwieldy. So to save time and space, we introduce a notational system of representing the power of number with a superscript as follows:

42 = 4 squared = 4 x 4 = 4 raised to the power 2

N3= N x N x N = N raised to the power 3


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3. Multiplication

The representation of numbers as powers of some base number simplifies the processes of multiplication and division.

Take the product 4 x 8 = 32. We could write this as

2x2 x 2x2x2 = 2x2x2x2x2

or

2 raised to the power 2
times
2 raised to the power 3
is
2 raised to the power 5

or

22 x 23 = 25

We can write this as a general rule, along the lines of

N raised to the power 'a'
times
N raised to the power 'b'
equals
N raised to the power 'a+b'

or

Multiplication Rule for Powers

Na x Nb = Na+b

Remember This!

Which we can read as: the product of a number raised to the power 'a' times the same number raised to the power 'b' is just the number raised to the power 'a+b'.

Here is another example:

Take the product 256 x 4 = 1024. This confirms:

4 raised to the power 4
times
4 raised to the power 1
is
4 raised to the power 5

To check that you understand, answer the following questions:

{ASK Number=2a,Number=2b,Number=2c}

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4. Division

The rule for division of numbers represented as powers is just as simple.

Take the division

81 ÷ 9 = 9

this is equivalent to

3 raised to the power 4
divided by
3 raised to the power 2
is
3 raised to the power 2

or

34 ÷ 32 = 32

You can see that dividing some power of a number N by another power of N is the same as subtracting the powers. We write this formally as:

Division Rule for Powers

Na ÷ Nb = Na-b

Remember This!

Here are some examples of divisions for you to try:

{ASK Number=3a,Number=3b,Number=3c}

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5. Zero and Negative Powers

We have established that when you have two numbers expressed as powers of some common base number (we have been using the symbol N to represent the base, e.g. in 32 the base is 3) then we can write a simple rule that gives us the power of the product (add the powers) and the power of the ratio (subtract the powers).

We can build on this to find two more general rules about powers.

First, let's consider what happens when you divide a number by itself:

9 ÷ 9 = 1

or in our power notation:

32 ÷ 32 = 1

But 32 ÷ 32 is just:

32-2 = 30

From which we can see the remarkable result that 30 is just 1. Of course we could have done this with any base, not just 3, so we get the general rule

N0 = 1

Any number raised to the power 0 is 1.

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Next, let's consider what happens when we divide a small number by a bigger number:

16 ÷ 64 = ¼

or in our power notation:

42 ÷ 43 = ¼

But 42 ÷ 43 is just:

42-3 = 4-1

From which we can see another remarkable result: that 4-1 is the same as ¼. Or that numbers less than 1 can be represented by negative powers!

1/Na=N-a

The reciprocal of N to the power 'a'

is N to the power '-a'

If you think about this for a while, it shouldn't seem so mysterious. After all, we know that multiplication and division are very closely related. Dividing by 4 is the same as multiplying by ¼. Or in powers: dividing by 41 is the same as multiplying by 4-1.

OK, check you understand by answering these questions on all we've done so far.

{ASK Number=5a,Number=5b,Number=5c,Number=5d,Number=5e}

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6. Powers of 10

Building on what we've learned so far, we will now concentrate on powers of ten only. Powers of ten are the building blocks for logarithms. So before we reveal what logarithms actually are, let's just explore some powers of 10:

Number
10x
Power
1000000
106
6
1000
103
3
100
102
2
10
101
1
1
100
0
0.1
10-1
-1
0.01
10-2
-2
0.001
10-3
-3
0.000001
10-6
-6

You should be able to see that there is a simple relationship between the power of ten and the number of zeros in the number. If the number is greater than 1, then the power of ten is just the number of zeros in the number: 1000 = 103. If the number is less than 1, then the power of ten is just minus the number of digits after the decimal point: 0.0001 = 10-4.

To check if you understand, try these simple sums:

{ASK Number=6a,Number=6b,Number=6c,Number=6d,Number=6e,Number=6f }

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7. Logarithms of Numbers which are Powers of 10

The logarithm of a number is simply the value of the power of 10 that equals the number. In the last section there was a table with 3 columns, and the last column, there called 'Power' could equally have been called 'Logarithm'.

So, since 100 is 102, then the logarithm of 100 is just 2. Since 0.001 is 10-3, then the logarithm of 0.001 is just -3.

There, that wasn't so bad was it!

Here is an equation that might help.

Definition of Logarithms

'y' is the logarithm of 'x' if 10y = x.

Remember this!

So 3 is the logarithm of 1000, because 103 is 1000.

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Aside: We have defined logarithms in terms of powers of 10. In fact we could have chosen powers of any number: we just would have changed the equation in the box above. So strictly speaking we have defined 'logarithms base 10' - that is logarithms using powers of 10. These are by far the most common. You will also see 'natural' logarithms, or logarithms base 'e'. These are just logarithms using powers of a number (called 'e') that is different from 10. Don't worry about this, we will work solely with powers of 10 from now on.

We will look in more detail at multiplication and division using logarithms in section 9, but just as a taster look at the following sum:

105 x 104 = 109

You should be able to see that the logarithm of the answer (9) is just the sum of the logarithms of the two numbers being multiplied together (5 and 4). This is just the rule for Multiplication of Powers that you remember from section 3. You should start to see that if we add the logarithms of two numbers we get the logarithm of the product of those two numbers.

Well, we'll go into this in more detail in a minute. First we will consider a much more important question: what is the logarithm of 2?


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8. Logarithms of Numbers Other Than Powers of 10.

We shall list some small powers of 10 and then draw a graph of them:

Number
Power of 10
Logarithm
0.01
10-2
-2
0.1
10-1
-1
1
100
0
10
101
1
100
102
2

From this graph, you should be able to see that the logarithms in the table are just some of the points on a continuous graph that covers all positive numbers. Thus for any positive number, we should be able to use the line on the graph to find its logarithm.

On the graph, look at the region on the x-axis between 1 and 10. You can see that the logarithm of 1 is 0, and that the logarithm of 10 is 1. You can also see that any number between 1 and 10 will have a logarithm between 0 and 1! Similarly, any number between 10 and 100 will have a logarithm between 1 and 2, and so on.

You should be able to see that we could use this graph for finding logarithms except for one thing - the x-axis is not a simple increasing set of values: each step seems to go up by a factor of 10. This means it is not easy to see how to find the right position on the x axis for any specific number. Normally, when we want to find a logarithm of a number other than a power of ten, we use a calculator to look up the logarithm on such a graph or a 'function' built into the calculator.

We will go through a simple arithmetic sum which will give us an approximate logarithm for 2, then list some logarithms of some useful numbers.

To find the logarithm of 2, we need to know 'y' in the expression 10y = 2. However we can use the following fact to help us: '2 to the power 10 is 1024'. That is 210 = 1024. We can now take this and note that 1024 is close to 1000, which is 103. So we can now write out in full:

2x2x2x2x2x2x2x2x2x2 is approximately 103

or

10yx10yx10yx10yx10yx10yx10yx10yx10yx10y is approximately 103

or

1010y is approximately 103

or

10y is approximately 3

or

y is approximately 0.3

I hope you can follow that. It gives us the useful result that the logarithm of 2 is about 0.3. That is 100.3 = 2. The actual value is about 0.301, so we were very close.

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Let's recap.

We have established that logarithms are just another name for a power of ten. We say that 100 is '10 to the power 2', but we can equally say the logarithm of 100 is 2. Similarly when we make a statement such as 'the logarithm of 2 is 0.3' what we mean is that 100.3 is equal to 2. The only difficult idea is fractional powers of ten. But then 100 is 1, and 101 is 10, so it should seem pretty unremarkable that 100.3 is a number like 2.

What are the logarithms for other small numbers? I'll give you a little table of them in the next section, but first let's see if we can calculate some of them from what we know already.

In particular we know the logarithm of 1, 2 and 10 (they're 0.0, 0.3 and 1.0), so how can we find the logarithm of, say, 4?

We can find the answer very easily by observing the parallels between these two sums:

2 x 2 = ?

100.3 x 100.3 = 10?

The answer to the first sum is 4. The answer to the second sum is, I hope you see, 100.6. Yes, we just use the rule for adding powers as we did at the start of the tutorial. But look! 100.3 is 2 ! The two sums are the same sums! The left hand sides are the same, so the right hand sides are the same, so 4 is equal to 100.6, so the logarithm of 4 is just 0.6.

Let's try one more, a little more tricky. What is the logarithm of 5?

Again, let us draw a parallel between two sums:

2 x ? = 10

100.3 x 10? = 101

The ? in the first sum must stand for 5. The ? in the second sum must stand for 0.7 (since we have a powers of 10 multiplication and the powers must add: 0.3+0.7=1.0). Once more the two sums are really the same, 2 is 100.3, 10 is 101, therefore 5 is 100.7, therefore the logarithm of 5 is 0.7.

OK. Now you do one.

{ASK Number=8}

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Here is a table of the approximate values of the logarithms of the whole numbers between 1 and 10.

Number
Approximate Logarithm
1
0
2
0.3
3
0.48
4
0.6
5
0.7
6
0.78
7
0.85
8
0.9
9
0.96
10
1

You should be able to see some patterns in this table. We've already seen that the logarithm for 4 is twice the logarithm for 2, and that the logarithm of 8 is the sum of the logarithm of 2 and the logarithm for 4.

So:

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There seems to be a general rule here:

We can always find the logarithm of a number by adding together the logarithms of its factors

That is if a number can be broken down into a multiplication sum of smaller numbers (we call these factors of the number), then we can find the logarithm of the number from the sum of the logarithms of the factors.

Thus using this table we can readily estimate the logarithm of any positive number that has all its factors in the table. We find entries in our table which when multiplied together form the number, then add together the logarithms of those entries.

Let's take the number 30. This can be factored as 5x6 or as 3x10. Using our rule that the logarithm of the product is the sum of the logarithms of the factors, we find that the logarithm of 30 is just 0.7+0.78 (using 5x6) or 0.48+1 (using 3x10); in either case the answer is 1.48. The logarithm of 30 is 1.48, i.e. 101.48 = 30.

Now it is your turn to estimate some logarithms.

{ASK Number=8a,Number=8b,Number=8c,Number=8d,Number=8e}

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9. Multiplication

What we have been doing in using our table of the logarithms of the numbers 1 to 10 to find the logarithms of other numbers has really been just a case of doing multiplication sums with logarithms. Let me show you why.

Here is a crazy way to multiply 2 by 5:

  1. Take the logarithm of 2 (=0.3)
  2. Take the logarithm of 5 (=0.7)
  3. Add the logarithms together (=1.0)
  4. Calculate what 10 is when raised to that power (101 = 10)
  5. That's the product of 2 and 5. (=10)

The reason this procedure seems crazy is that the sum 2x5 is one we can do in our head anyway.

However let's do the same for the sum 1.4729 x 3.6128:

  1. Take the logarithm of 1.4729 (=0.1682, so my calculator says)
  2. Take the logarithm of 3.6128 (=0.5584)
  3. Add the logarithms together (=0.7266)
  4. Calculate what 10 is raised to that power (100.7266 = 5.3290)
  5. That's the product of 1.4729 and 3.6128 (=5.3290)

Hey! We were able to do the multiplication by using 3 simple operations: finding the logarithm of a number, adding logarithms, finding the value of a power of 10. This last operation (step 4) is sometimes given the horrible name of the 'antilogarithm' calculation. Life would have been simpler if the first operation was called 'anti-powers-of-ten' instead of 'logarithm'.

So here's the general rule for doing a multiplication using logarithms:

The product of numbers A and B is just the antilogarithm of the sum of the logarithm of A and the logarithm of B:

A x B = 10(log(A)+log(B))

This looks quite unmemorable, so just remember the steps in the calculation above. To multiply two numbers together using logarithms, first take the logarithm of each of the numbers, then add these logarithms together, then take the antilogarithm (that is the power of ten) of the sum, the result is the product of the two numbers.

Here are some simple multiplication sums for you to try. You shouldn't need a calculator.

{ASK Number=9a,Number=9b,Number=9c}

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10. Division

This is the last section of the tutorial. You have done well to get this far.

This section is about using logarithms to divide numbers. Don't worry that this might somehow be more complicated than multiplying numbers with logarithms. Just the reverse - in fact we do pretty much the same steps as for multiplication but we subtract the logarithms instead of adding them.

If we want to multiply A by B, remember, we take logarithms of these numbers (this gives us: log(A) and log(B)), then add the logarithms (to get: log(A)+log(B)) then undo the logarithm by taking the power of ten.

Similarly if we want to divide A by B, we take logarithms of these numbers (to give us: log(A) and log(B)), then subtract the logarithm of B from the logarithm of A (to get: log(A) - log(B)), and lastly undo the logarithm by taking the power of ten as before.

Let us go through a worked example. Let us say we want to divide 15 by 5. (and let us say we want to do it the hard way!):

  1. Take the logarithm of 15: log(15)=1.18
  2. Take the logarithm of 5: log(5)=0.7
  3. Subtract the logarithm of 5 from the logarithm of 15: log(15)-log(5) = 1.18 - 0.7 = 0.48
  4. Undo the logarithm by taking the power of ten: 100.48 = 3

If you find this hard, try to remember that logarithms are just a shorthand for powers of ten. When we are dividing 15 by 5 this way, really we are just doing this sum with powers of ten:

15 ÷ 5 = 101.18 ÷ 100.7 = 100.48 = 3

Here's a summary:

The division of number A by B is just the antilogarithm of
(the logarithm of A minus the logarithm of B):

A ÷ B = 10(log(A)-log(B))

And here's one last set of questions to try:

{ASK Number=10a,Number=10b,Number=10c}

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That's it!

You answered {nanswer} questions out of {nquestion}. You got a score of {nscore} out of a possible {ntotal} or {npercent}% right. Well done!

Finally check you really can do the following:

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© 1998 Mark Huckvale University College London

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