CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!


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1 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 5 2 or 4 3. 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: 3 2. 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 3 2, 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. 3 2 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, 3 2 will always mean 3 x 3 and give the answer 9. A further example: 4 3. 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, 5 4 is read as five [raised] to the power [of] 4, or five [raised] to the 4 th [power]. Another example: 2 4 (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
2 Find the value of each of the following. Also, for each question, work out which numbers represent the base and the power 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 3 2, 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 , 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. Find the value of each of the following. 1. (4) 2 2. (3) (5) Again you can check these results on your calculator and the answers and explanations are at the end. Fractions For example, ( 3 4 )2 is the same as , = (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
3 Operations with powers. Let s bring in a little bit of Algebra here. Now don t worry, Algebra simply generalises what happens to numbers. So, for example, a 2 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 want to calculate a 2 x a 3. We could write this out longhand, and obtain a x a x a x a x a, which is a 5. 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. a p x a q = a p+q Example: Find the value of 2 5 x 2 4. Shortcut method: 2 5 x 2 4 = = 2 9. (Longer method: 2 5 x 2 4 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 9 ). 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: = = = Did you think subtract? 3
4 This gives us our second general rule for operations with powers: a p a q = a pq Write your answers to these questions using power notation: x x Combinations For example, (2 3 ) 4. This would mean 2 3 x 2 3 x 2 3 x 2 3. If we add the powers, we would end up with (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 2 12!) 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: (a p ) q = a p q or we can also write (a p ) q = a pq (By the way, a number raised to the power 1 is just the number itself. Examples 3 1 = 3, = 752 etc.) Write your answer as a power in each case. 1. (3 2 ) 4 2. (2 5 ) 2 3. (4 1 ) 3 4. (10 2 ) 3 Zero power Taking the division rule a step further: what happens if we have ? Using what we know about squaring, 5 2 = 25, so we would get 25 25, which is 1. But what about using our other methods? 4
5 Firstly, using the subtraction of powers: = = 5 0 Now using division: = 1 So, 5 0 is the same as 1. This goes for any number, so we can generalise again and write: a 0 = 1 This is the 3 rd rule for indices/powers. Here are some for you to try: x (3 x 5) a 0 Negative powers And further than 0: let s say we have to calculate 3 4 subtraction of powers, (3 45 ), we would end up with Using the Using division, we would get = = 1 3 This means that 31 is the same as 1 3. If we had , we would end up with 34 or with 1 34, and this all leads to the rule : a p = 1 a p 5
6 Another example: 62 = = 1 36 Write each answer with a positive power: 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 10 2 = 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: log = 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 10 2 = 100 means exactly the same as log = 2. Another example: 2 5 = 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 ). Further example: 31 = 1 3. So, log 3( 1 3 ) = 1 Write each power statement in its log[arithm] form: = = = = 1 9 6
7 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: log 2 8. So you need to work out the power to which 2 is raised to get 8. Solution: 2 x 2 x 2 = 8 so 2 3 = 8,which means that the log value is 3. So log 2 8 = 3. What about log ? We need to remember our place value, and that 0.1 is the same as 1, 10 which means the same as So, log is 1. You can actually check any log 10 on the calculator. Just type log 0.1 and you should see 1. (Note that on the calculator, log stands for log 10. 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.) Find the value of the log in each case: 1. log log log log log log 5 ( 1 25 ) 7. log 2 ( 1 8 ) 8. log 10( ) 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: log b 9 = 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 3 2 = 9, which means b = 3. (Notice that for b 2 = 9, b could also be 3, however we don t use negative bases with logs.) Find the value of the base (b). 1. log b 16 = 2 2. log b 81 = 4 3. log b = log b (½) = 1 7
8 Now, find the value of the number. Let s call the number n. Here is an example: log 2 n = 3. This means that 2 3 = n, so n = 8. Find the value of n. 1. log 2 n = 5 2. log 3 n = 4 3. log 2 n = log 4 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. In fact we can even write a a = a, where a represents any number. But what about this? = = 2 1 = 2 Because we are multiplying 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 And 3 is the same as and so on. So we can write: 8
9 a 1 2 = a (Note: must not be confused with 2 1! The first is 2 raised to the power ½, 2 the second is read as two and a half.) As well, a = a (the cubed root of a), and a = a (the 4 th root of a). 3 (For example, 8 is 2 because we look for the number, which, when cubed gives 8. And so, = 2, for the same reason!) What about 8 2 3?! We can think of this in two different ways. Either, we can use (8 2 ) , which means we would look for 64 3 = 64 = 4, or we can use (8 1 3 ) 2 3, which means we would look for ( 8 ) 2 = 2 2 = 4. Either method works just as well (though the simpler numbers are probably easier). Another example: I m going to use ( ), because the numbers are simpler = 4 and 4 3 = 64. So = 64. (Notice we always use improper fractions for powers and do not use mixed numbers, so we use 3 instead of 1½.) 2 Again we can generalize this as a rule: p q a q = a p q or ( a) p Find the value of each of the following:
10 (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 4 2, we get 16, which is bigger than 8. But if we do 4 1, 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 4 x = 8. Now, here s the trick. 4 is the same as 2 2, and 8 is the same as 2 3. So, we have: (2 2 ) x = 2 3. We can multiply the powers and get 2 2x = is the same base on both sides, so the powers (2x and 3) must be the same, so 2x = 3. This tells us that x = 3 2. So, log 4 8 is 3 2. (You can check this by using your calculator, and finding if is 8.) Here is another example, with just the procedure set out. Find the value of log Let x = log x = 243 (3 2 ) x =
11 3 2x = 3 5 2x = 5 x = 5 2 log = 5 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) , or Level 3 (top floor), Building 11, or through your campus. 11
12 Answers (From page 2) is 8 (it is 2 x 2 x 2) = 81 (it is 3 x 3 x 3 x 3) = 100 (it is 10 x 10) = 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) = 10 x 10 x 10 5 x 5 x 5 = = (5) 3 = 10 x 10 x x 5 x 5 = = 875 (note 3 and 4 are the same) = 10 x 10 4 x 4 = = 84 Operations with powers (from page 4) x 2 5 = = = = x = = 5 5 (or you can do = = = 5 5 ) Combinations (from page 4) 1. (3 2 ) 4 = (2 5 ) 2 = (4 1 ) 3 = (10 2 ) 3 = 10 6 Zero power (from page 5) = = x 3 0 = 4 x 1 = = = 5 3 or = = (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 a 0 means 4 x a 0 = 4 x 1 = 4 Negative powers(from page 6) = = = 43 =
13 Logarithms (from page 6) Writing in log form = 25 is the same as log 5 25 = = 64 is the same as log 4 64 = = 64 is the same as log 8 64 = = 1 9 is the same as log = 2 Finding the value of the log 1. log 2 4 = 2 because 2 2 = 4, so the answer (the logarithm, or power) is log = 3 because 10 3 = 1000, so the answer (the logarithm, or power) is log 3 81= 4 because 3 4 = 81, so the answer is log = 2, because 102 = 1 100, so the answer is log 5 25 = 2, because 5 2 = 25, so the answer is 2. 6.log 5 ( 1 ) = 2, because 52 = 1, so the answer is log 2 ( 1 ) 8 = 3, because 23 = 1, so the answer is log 10 ( 1 ) 1000 = 3, because 103 = 1, so the answer is Finding the base: 1. log b 16 = 2. This means b 2 = 16, so b = log b 81 = 4. This means b 2 = 81, so b = log b = 3. This means b 3 = 0.001, so b =
14 4. log b (½) = 1. This means b 1 = ½, so b = 2. Finding the number: 1. log 2 n = 5. This means 2 5 = n, so n = log 3 n = 4. This means 3 4 = n, so n = log 2 n = 3. This means 23 = n, so n = ⅛. 4. log 4 n = 1. This means 41 = n, so n = ¼. Last rule for powers (from page 9) = 25 = = 81 = = ( 49) 3 = 7 3 = = ( 125) 2 = 5 2 = 25 14
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