Download CONNECT: Powers and logs

CONNECT: Powers and logs
POWERS, INDICES, EXPONENTS, LOGARITHMS – THEY ARE ALL
THE SAME!
You may have come across the terms powers, indices, exponents and
logarithms. But what do they mean?
The terms power(s), index (indices), exponent(s) in Mathematics are actually
interchangeable. All of them are that little number written above, and to the
right of, another number, such as 52 or 43. Some of those little numbers
(written as superscripts) have special names. You are probably familiar with
squaring and cubing a number. But let’s start at the beginning!
We might have to calculate 3 x 3. Rather than write out both those 3s, we
use a shorthand notation: 32. The superscript 2 tells us that 3 is to be
multiplied by itself, and we would get the answer 9. Note: the result is not 6
although there are two 3s, because two 3s would be 2 x 3, or 3 + 3, not 3 x 3.
In the example 32, 3 is called the base and 2 is called the power (or index or
exponent). We’ll use power from now on, but remember that we can just as
easily write index or exponent. 32 is read as three raised to the power of two,
or simply three to the power two. More commonly, when the power is 2, we
use the word squared, so we can also read this as three squared. No matter
which way we express it, 32 will always mean 3 x 3 and give the answer 9.
A further example: 43. This is read as four raised to the power of three, or
four to the power three, or four cubed. It means 4 x 4 x 4 and will give the
result 64 because 4 x 4 = 16 and 16 x 4 = 64. The base in this case is 4 and
the power is 3.
By the way, the powers 2 and 3 are the only ones that have special names.
So, for example, 54 is read as five [raised] to the power [of] 4, or five [raised]
to the 4th [power].
Another example: 24 (read two to the power four) is 2 x 2 x 2 x 2, which
makes 16. Here, the base is 2 and the power is 4. (Note how efficient the
notation is – we don’t have to write out all those 2’s!)
(Although it might seem trivial naming the base and power, they are important
items of vocabulary for when we use logarithms – we’ll get to this later!)
Over the page are some for you to try.
1
Find the value of each of the following. Also, for each question, work
out which numbers represent the base and the power.
1.
23
2. 34
3. 102
4. 53
You can check these results on your calculator. (Also, answers and
explanations are provided at the end of this resource). If you are not sure
how to use your calculator, you can have a look at CONNECT: Calculators –
GETTING TO KNOW YOUR SCIENTIFIC CALCULATOR.
Raising a negative number to a power.
Let’s say we have to raise -3 to the power 2. This MUST be written as (-3)2.
The reason for this is that we need to multiply -3 x -3. If we write -32, without
the brackets, this implies that we square the 3 first (because of the Order of
Operations), then put a minus sign in front of the answer! (It is similar to
doing 15 – 32, say, which is the same as 15 – 9 and gives 6.) The correct
answer to raising -3 to the power 2 is 9. If you SQUARE ANY number,
positive or negative, you will ALWAYS get a POSITIVE result.
What about (-2)3? This means -2 x -2 x -2, and gives the result -8 (because -2
x -2 = 4 and 4 x -2 = -8.) Notice this time, when we cube a negative number,
we obtain a NEGATIVE result.
Here are some for you to try.
Find the value of each of the following.
1. (-4)2
2. (-3)4
3. 103 – 53
4. 103 + (-5)3
5. 102 – 42
Again you can check these results on your calculator and the answers and
explanations are at the end.
Fractions
3
3
3
9
For example, ( )2 is the same as × , = .
4
4
4
16
(Remember, when multiplying fractions, multiply across numerators and
across denominators. If you are not sure how to, you can refer to CONNECT:
Fractions. FRACTIONS 2 – OPERATIONS WITH FRACTIONS: x and ÷
2
Operations with powers.
Let’s bring in a little bit of Algebra here. Now don’t worry, Algebra simply
2
generalises what happens to numbers. So, for example, a just means a x a,
where a is the base, 2 is the power and a can represent any number.
There are some shortcuts to working out calculations with powers. Say we
2
3
want to calculate a x a . We could write this out longhand, and obtain a x a x
a x a x a, which is a5. Notice that the power, 5, is also the result of adding the
powers 2 and 3. This happens in every case. So, to multiply two powers of
the same base, just add the powers.
This is our first general “rule” for operations with powers. We can use letters
for the powers as well, but remember the letters simply stand for the general
case and you can use the rule every time you recognise it.
ap x aq = ap+q
Example: Find the value of 25 x 24.
Shortcut method: 25 x 24 = 25+4 = 29.
(Longer method: 25 x 24 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 29).
Note that the base numbers must be the same for this “rule” to work.
Now, we’ve just seen that when you multiply the same base number raised to
powers that you can actually add the powers. So it follows that if you are
dividing the same base number raised to powers, then you would _________
the powers 1.
We can illustrate this as follows: 36 ÷ 32 =
=
3×3×3×3×3×3
3 × 3×3×3
= 34
1
Did you think subtract?
3
3×3
1
This gives us our second general “rule” for operations with powers:
ap ÷ aq = ap-q
Here are some for you to try.
Write your answers to these questions using power notation:
1. 23 x 25
2. 38 ÷ 34
3. 54 x 53 ÷ 52
Combinations
For example, (23)4. This would mean 23 x 23 x 23 x 23. If we add the powers,
we would end up with 212. (And if we wrote out 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
x 2 x 2 x 2, we would still get 212!) Notice we could have obtained the same
result for the power if we had multiplied the 3 x 4. This is the same in every
case. So, we can write:
(𝑎𝑝 )𝑞 = 𝑎𝑝×𝑞 or we can also write (𝑎𝑝 )𝑞 = 𝑎𝑝𝑞
(By the way, a number raised to the power 1 is just the number itself.
Examples 31 = 3, 7521 = 752 etc.)
Here are some for you to try.
Write your answer as a power in each case.
1. (32)4
2. (25)2
3. (41)3
4. (102)3
Zero power
Taking the division rule a step further: what happens if we have 52 ÷ 52?
Using what we know about squaring, 52 = 25, so we would get 25 ÷ 25, which
is 1. But what about using our other methods?
4
Firstly, using the subtraction of powers: 52 ÷ 52 = 52-2 = 50
Now using division:
52
52
=1
So, 50 is the same as 1. This goes for any number, so we can generalise
again and write:
a0 = 1
This is the 3rd “rule” for indices/powers.
Here are some for you to try:
1. 100
2. 3840
5. (3 x 5)0
6. 4a0
3. 4 x 30
4. 53 ÷ 50
Negative powers
And further than 0: let’s say we have to calculate 34 ÷ 35.
subtraction of powers, (34-5), we would end up with 3-1.
Using the
Using division, we would get
34 ÷ 35 =
=
3×3×3×3
3×3×3×3×3
1
3
1
This means that 3-1 is the same as .
2
6
3
If we had 3 ÷ 3 , we would end up with 3-4 or with
“rule”:
𝑎−𝑝 =
5
1
𝑎𝑝
1
34
, and this all leads to the
Another example: 6-2 =
1
62
=
1
36
Here are some for you to try.
Write each answer with a positive power:
1. 2-5
2. 7-3
3. 42 ÷ 45
There is one more “rule” to deal with and we will do that on page 8. But first,
let’s discuss logarithms.
Logarithms
Although these really are the same as powers, we write them slightly
differently.
The easiest way to show you how logarithms work is by an example.
(Note: we use log to stand for logarithm.)
We know that 102 = 100. Now, let’s try to put the 2 by itself and the 10 and
the 100 on the other side of the equals sign. That’s what logarithms do. A
logarithm is a power. We write: log10 100 = 2. This is the shorthand that tells
us that 2 is the power (or logarithm) to which we raise the base 10 to get the
number 100. We say this as: “the log[arithm] to base 10 of 100 is equal to 2”.
So 102 = 100 means exactly the same as log10 100 = 2.
Another example: 25 = 32. So log 2 32 = 5, because 5 is the power to which
we raise 2 to get 32. We read “log to base 2 of 32 is equal to 5” (or
sometimes “log of 32 to base 2 is 5”).
1
1
Further example: 3-1 = . So, log 3 ( ) = −1
3
3
Here are some for you to try.
Write each power statement in its log[arithm] form:
1. 52 = 25
6
2. 43 = 64
3. 82 = 64
4. 3-2 =
1
9
Now that you’ve mastered power and log form, the next trick is to work out
parts of logarithm expressions, such as the base, the number, or the log itself.
For example, you might be asked to find the value of this log: log28. So you
need to work out the power to which 2 is raised to get 8.
Solution: 2 x 2 x 2 = 8 so 23 = 8,which means that the log value is 3.
So log28 = 3.
What about log100.1? We need to remember our place value, and that 0.1 is
the same as
1
, which means the same as 10-1. So, log100.1 is -1.
10
You can actually check any log10 on the calculator. Just type log 0.1 and you
should see -1. (Note that on the calculator, log stands for “log10”. You don’t
need to type the 10 into the calculator. Unfortunately, however, 10 is the only
base for logs on the calculator, apart from the special case of logs to base e,
which you may work with later.)
Here are some for you to try.
Find the value of the log in each case:
1. log24
2. log101000
5. log525
6.log5(25)
1
3. log381
1
7. log2(8)
4. log100.01
8. log10(
1
1000
)
Now, find the value of the base. Let’s call the base b, but only because we
don’t know its value yet. Here is an example: logb9 = 2. So, we need to
know what number squared is 9. That is, we need to find b for 𝑏 2 = 9. The
base is 3 because 32 = 9, which means b = 3.
(Notice that for 𝑏 2 = 9,
bases with logs.)
b could also be –3, however we don’t use negative
Here are some for you to try.
Find the value of the base (b).
7
1. logb16 = 2
2. logb81 = 4
3. logb0.001 = -3
4. logb(½) = -1
Now, find the value of the number. Let’s call the number n.
example: log2 n = 3. This means that 23 = n, so n = 8.
Here is an
Here are some for you to try.
Find the value of n.
1. log2 n = 5
2. log3 n = 4
3. log2 n = -3
4. log4 n = -1
Last rule for powers
Suppose we multiply √2 × √2.
We must get 2, because that is what √2 means.
In the same way, √3 × √3 = 3 or √185 × √185 = 185.
In fact we can even write √𝑎 × √𝑎 = 𝑎, where a represents any number.
But what about this?
1
1
22 × 22
1 1
= 2 2 +2
= 21
=2
1
Because we are multiplying 22 by itself, and ended up with 2, just as I
multiplied √2 by itself, and ended up with 2, it follows that √2 is the same as
1
1
22 . And √3 is the same as 32 and so on.
So we can write:
8
1
𝑎 2 = √𝑎
1
1
(Note: 22 must not be confused with 2 ! The first is 2 raised to the power ½,
2
the second is read as “two and a half”.)
1
1
3
4
As well, 𝑎3 = √𝑎 (the cubed root of 𝑎), and 𝑎4 = √𝑎 (the 4th root of 𝑎).
3
(For example, √8 is 2 because we look for the number, which, when cubed
1
gives 8. And so, 83 = 2, for the same reason!)
2
What about 83 ?! We can think of this in two different ways. Either, we can
1
1
3
use (82 ) 3 , which means we would look for 643 = √64 = 4, or we can use
1
(83 )2 , which means we would look for ( 3√8)2 = 22 = 4. Either method works
just as well (though the simpler numbers are probably easier).
3
2
1 3
2 ,
Another example: 16 . I’m going to use (16 )
1
2
3
2
simpler. 16 = 4 and 43 = 64. So 16 = 64.
because the numbers are
(Notice we always use improper fractions for powers and do not use mixed
numbers, so we use
3
2
instead of 1½.)
Again we can generalize this as a rule:
𝑝
𝑞
𝑞
𝑎𝑞 = √𝑎𝑝 or ( √𝑎)p
Here are some for you to try.
Find the value of each of the following:
1. 25
9
1
2
2. 81
1
4
3. 49
3
2
4. 125
2
3
(You can check your answers on your calculator. If you are not sure how to
do this, please refer to CONNECT: Calculators: GETTING TO KNOW YOUR
SCIENTIFIC CALCULATOR.)
Using fraction powers with logs
This section puts it all together!
Example: find the value of log 4 8.
This means we need to know the power to which 4 is raised to get 8. Many
people write the answer 2 here because 4 x 2 is 8. But remember, we are
looking for 4 raised to a power, that is 4 x 4 x …, not 4 x 2.
Back to finding log 4 8.
To what power can I raise 4, to get 8? This is a bit tough, because, if we do
42, we get 16, which is bigger than 8. But if we do 41, we only get 4. So our
power must be somewhere between 1 and 2.
What we can do is: let x be the value of the log we are looking for, that is, let
x = log 4 8. If we put this into power form, we would get 4x = 8.
Now, here’s the trick. 4 is the same as 22, and 8 is the same as 23. So, we
x
x
have: (22) = 23. We can multiply the powers and get 22 = 23.
2 is the same base on both sides, so the powers (2x and 3) must be the same,
3
3
so 2x = 3. This tells us that x = 2. So, log 4 8 is 2.
3
(You can check this by using your calculator, and finding if 42 is 8.)
Here is another example, with just the procedure set out.
Find the value of log 9 243
Let x = log 9 243
∴ 9𝑥 = 243
(32 )𝑥 = 35
10
32𝑥 = 35
2𝑥 = 5
𝑥=
5
2
5
∴ log 9 243 = 2
Last of all, there are some rules which we can look at for logarithms as well.
We’ll do them in CONNECT:Powers and logs2, where we will also look at
some uses of logarithms.
If you need help with any of the Maths covered in this resource (or any other
Maths topics), you can make an appointment with Learning Development
through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11,
or through your campus.
11
Answers
(From page 2)
1. 23 is 8 (it is 2 x 2 x 2).
2. 34 = 81 (it is 3 x 3 x 3 x 3).
3. 102 = 100 (it is 10 x 10).
4. 53 = 125 (it is 5 x 5 x 5).
Raising a negative number to a power (from page 2)
1. (-4)2 = 16 (it is -4 x -4).
2. (-3)4 = 81 (it is -3 x -3 x -3 x -3).
3. 103 – 53 = 10 x 10 x 10 – 5 x 5 x 5 = 1000 – 125 = 875
4. 103 + (-5)3 = 10 x 10 x 10 + -5 x -5 x -5 = 1000 + -125 = 875
(note 3 and 4 are the same)
5. 102 – 42 = 10 x 10 – 4 x 4 = 100 – 16 = 84
Operations with powers (from page 4)
1. 23 x 25 = 23+5 = 28
2. 38 ÷ 34 = 38-4 = 34
3. 54 x 53 ÷ 52 = 54+3-2 = 55 (or you can do 54+3 ÷ 52 = 57 ÷ 52 = 57-2 = 55)
Combinations (from page 4)
1. (32)4 = 38
2. (25)2 = 210
3. (41)3 = 43
4. (102)3 = 106
Zero power (from page 5)
1. 100 = 1
2. 3840 = 1
3. 4 x 30 = 4 x 1 = 4
4. 53 ÷ 50 = 53-0 = 53 or 53 ÷ 50 = 53 ÷ 1 = 53
5. (3 x 5)0 = 1 (you don’t even have to worry about doing the inside of the
brackets first here – the 0 index tells you straight away that your result
is 1.
0
0
6. 4a means 4 x a = 4 x 1 = 4
Negative powers(from page 6)
1. 2-5 =
12
1
25
2. 7-3 =
1
73
3. 42 ÷ 45 = 4-3 =
1
43
Logarithms (from page 6)
Writing in log form
1. 52 = 25 is the same as log 5 25 = 2
2. 43 = 64 is the same as log 4 64 = 3
3. 82 = 64 is the same as log 8 64 = 2
4. 3-2 =
1
9
1
is the same as log 3 9 = −2
Finding the value of the log
1. log24 = 2 because 22 = 4, so the answer (the logarithm, or power) is
2.
2. log101000 = 3 because 103 = 1000, so the answer (the logarithm, or
power) is 3.
3. log381= 4 because 34 = 81, so the answer is 4.
4. log100.01= -2, because 10-2 =
1
100
, so the answer is -2
5. log525 = 2, because 52 = 25, so the answer is 2.
1
1
6.log5(25) = -2, because 5-2 = 25, so the answer is -2.
1
1
7. log2( ) = -3, because 2-3 = , so the answer is -3.
8
8. log10(
1
1000
8
1
) = -3, because 10-3 = 1000, so the answer is -3.
Finding the base:
1. logb16 = 2. This means 𝑏 2 = 16, so b = 4.
2. logb81 = 4. This means 𝑏 2 = 81, so b = 9.
3. logb0.001 = -3. This means 𝑏 −3 = 0.001, so b = 10.
13
4. logb(½) = -1. This means 𝑏 −1 = ½, so b = 2.
Finding the number:
1. log2 n = 5. This means 25 = n, so n = 32.
2. log3 n = 4. This means 34 = n, so n = 81.
3. log2 n = -3. This means 2-3 = n, so n = ⅛.
4. log4 n = -1. This means 4-1 = n, so n = ¼.
Last rule for powers (from page 9)
1
1. 252 = √25 = 5
1
4
2. 814 = √81 = 3
3
3. 492 = (√49)3 = 73 = 343
2
3
4. 1253 = ( √125)2 = 52 = 25
14