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CONNECT: Powers and Logs POWERS, INDICES, EXPONENTS, LOGARITHMS – THEY ARE ALL the SAME!

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CONNECT: Powers and Logs POWERS, INDICES, EXPONENTS, LOGARITHMS – THEY ARE ALL the SAME! 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 For example, ( ) is the same as × , = . 3 3 3 9 2 (Remember, when4 multiplying fractions,4 4 16multiply 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 5 a x a x a, which is a . 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. p q p+q a x a = a 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 powers1. × × × × × We can illustrate this as follows: 3 ÷ 3 = × 6 2 3 3 3 3 3 3 × × 3× 3 = 3 3 3 3 = 3 1 4 1 Did you think subtract? 3 This gives us our second general “rule” for operations with powers: p q p-q a ÷ a = a 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: × ( ) = or we can also write ( ) = 푝 푞 푝 푞 푝 푞 푝푞 푎 푎 푎 푎 (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 = 1 Now using division: 2 5 2 So, 50 is the same as5 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 3. 4 x 30 4. 53 ÷ 50 5. (3 x 5)0 6. 4a0 Negative powers And further than 0: let’s say we have to calculate 34 ÷ 35. Using the subtraction of powers, (34-5), we would end up with 3-1. Using division, we would get × × × 3 ÷ 3 = × × × × 3 3 3 3 4 5 3 3 3 3 3 = 1 3 This means that 3-1 is the same as . 1 2 6 -4 If we had 3 ÷ 3 , we would end up3 with 3 or with , and this all leads to the 1 “rule”: 4 3 1 = −푝 푎 푝 푎 5 -2 Another example: 6 = = 1 1 2 6 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: log 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 10 number 100.
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