Read an Excerpt
Attacking Problems in Logarithms and Exponential Functions
By David S. Kahn Dover Publications, Inc.
Copyright © 2015 David S. Kahn
All rights reserved.
ISBN: 978-0-486-80801-7
CHAPTER 1
UNIT ONE
Seven Simple Rules for Working with Exponents
When we multiply a number by itself, we call this process squaring. For example, when we multiply 3·3 = 9 we use a short-hand method. We write the expression as 32, where the 2 indicates that we have multiplied the number by itself 2 times. So, if we wrote 33, that would indicate that we have multiplied the number by itself 3 times, that is, 3·3·3 = 27. Similarly, 34 = 3·3·3·3 = 81, and so on. Thus, 56 = 5·5·5·5·5·5 = 15,625. Got the idea?
Notice how quickly the result grows when we multiply a number by itself. For example, we only had to multiply 5 by itself 6 times to get 15,625. Think about what this means. If we had 5 people in a room, and each of them brought 5 friends into the room, we would have 52 = 25 people. If we now had each of those people bring in 5 friends, we would have 53 = 125 people. If we then repeated the process three more times, we would have 15,625 people. We would need a very big room! This is the essential fact of exponential growth, which we will explore in this book. Namely, that one can start with a small number and, in just a few steps, end up with a very large number.
Now let's figure out some rules about exponents. What does it mean to raise a number to the power of 1? Let's see. If we write 41, we multiply 4 one time. That is, 41 = 4. Now let's dispense with using numbers. If we write x1, we multiply x, one time, so x1 = x.
Now, what if we raise x to the power a. This means that we multiply x by itself a times.
Now suppose we have two expressions, x3 and x5. What happens when we multiply them together? Remember that x3 = x·x·x and x5 = x·x·x·x·x, so if we multiply them, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We have x multiplied by itself 8 times, so x3 · x5 = x8. Now let's do this again, but this time with xa and xb. If we multiply them, we get xa · xb = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This is our second rule: xa · xb = xa+b.
Notice that this requires that the two expressions have the same base, namely x. What if you have xa · yb? You can't do anything with this expression. For example, if you have 53 · 42, you have to evaluate each separately to get 53 = 125 and 42 = 16, and thus 53 · 42 = 125 · 16 = 2000.
By the way, if you have 52 · 42 instead, you could evaluate them separately: 52 = 25 and 42 = 16. So 52 · 42 = 25 · 16 = 400, or you could combine them into (5 · 4)2 = 202 = 400. This means that xa · ya = (xy)a. Be careful when you combine expressions with different bases!
Suppose that, instead of multiplying x5 and x3, we divide them. We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The x's in the numerator and denominator will cancel, leaving us with x5/x3 = x5-3 = x2. Now let's do this again but with xa and xb. We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The x's in the numerator and denominator will cancel, leaving us with xa/xb = xa-b. (Note that x cannot be zero.)
This our third rule: xa/xb = xa-b.
What if a and b are the same number? Let's divide x5 and x5. We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This time, all of the x's in the numerator and denominator will cancel, leaving us with x5/x5 = x5-5 = x0 = 1. Now let's do this again but with xa and xa. We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The x's in the numerator and denominator will cancel, leaving us with xa/xa = xa-a = x0 = 1.
This is not a rule, but remember x0 = 1!
What does it mean to raise an exponent to an exponent? That is, what does it mean if we have an expression like (x2)3? This means that we have (x2)3 = x2 · x2 · x2. Well, using our rule for multiplying exponential expressions, we add the exponents and get (x2)3 = x2 · x2 · x2 = x6. Let's do this again, but with xa. We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If we add up the xa terms, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This is our fourth rule: (xa)b = xab.
It is important that you don't confuse multiplying terms where you add the exponents, with raising a term to a power, where you multiply the exponents.
What does it mean if we raise x to a negative exponent? Suppose we have x-4. We can think of this as x0-4. From our rule above, we know that this means x-4 = x0-4 = x0/x4. But we also know that x0 = 1, so x-4 = x0-4 = x0/x4 = 1/x4.
Let's do this again, but with xa. We get x-a = x0-a = x0/xa = 1/xa.
This is our fifth rule: x-a = 1/xa.
So far, we have seen what happens when we raise x to a positive number, a negative number and zero. What's left? Let's see what happens when we raise x to a fraction.
Suppose we want to evaluate 53. We know that 53= 5·5·5 = 125. If instead, we wanted to find a single power that we could raise 125 to in order to get 5, what could we do? We raised 5 to the power of 3 in order to get 125, so let's raise 125 to the 1/3 power. This gives us: 1251/3 = (53)1/3, and using our rules of exponents, we get 1251/3 = 53 1/3 = 51 = 5. Notice that this is the cube root! In other words, 1251/3 = [cube root of 125] = 5.
Now let's do this again, but with x1/a. Suppose we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let's call this the result of raising ya = x. Then, if we raise both sides to the power 1/a, we get (ya)1/a = x1/a. Using our rules of exponents, we get (ya)1/a = ya·[1/a] = y1 = x1/a. Also remember that we got x by finding ya = x, so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This is our sixth rule: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
What if we see x raised to a power like 5/3, such as with 85/3 Simple. We can think of this as (81/3)5. First we find 81/3, which we know is [cube root of 8] = 2. Then we can evaluate 25 = 32. In other words, 85/3 = ([cube root of 8])5. Of course, we could first have found (85)1/3. But who would want to find [cube root of 32768]?!
Now let's do this again, but with xb/a. We can think of this as (x1/a)b. We know from above that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so we rewrite this as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This gives us our seventh rule: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Here are our rules again:
Rule Number One: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Rule Number Two: xa · xb = xa+b
Rule Number Three: xa/xb = xa-b
Rule Number Four: (xa)b = xab
Rule Number Five: x-a = 1/xa
Rule Number Six: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Rule Number Seven: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
And remember that x0 = 1 and x1 = x!
Let's do some practice problems.
Practice Problem Set #1
Evaluate the following:
1 a) 61 = b) 62 = c) 60 = d) 64 =
2 a) 43 · 42 = b) 43/42 = c) (43)2 = d) (42)3 =
3 a) x5 · x9 = b) y3 · y4 = c) x · x5 = d) x3 · x5 · x10 =
4 a) a3b · a4b = b) ya+1 · ya-1 = c) z1+a · z1-a = d) x6 · x-6 =
5 a) x8/x3 = b) x5/x-5 = c) y3/y5 = d) z2/z-4 =
6 a) (x2)1 = b) (y4)-2 = c) (z-5)0 = d) (a8)1/2 =
7 a) [square root of y] = b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] d) [cube root of z-4] =
Solutions to Practice Problem Set #1
Evaluate the following:
1 a) 61 =
Any number raised to the power 1 is itself, so 61 = 6.
b) 62 =
Remember from Rule Number One that 62 means 6·6, so 62 = 6·6 = 36.
c) 60 =
Any number raised to the power 0 (except 0 itself) is 1, so 60 = 1.
d) 64 =
Remember from Rule Number One that 64 means 6·6·6·6, so 64 = 6·6·6·6 = 1296.
2 a) 43 · 42 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: 43 · 42 = 43+2 = 45 = 1024.
b) 43/42 =
Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: 43/42 = 43-2 =41 = 4.
c) (43)2 =
Rule Number Four says that when a number raised to a power is raised to a power, we multiply the powers. Here we get: (43)2 = 46 = 4096.
d) (42)3 =
Rule Number Four says that when a number raised to a power is raised to a power, we multiply the powers. We multiply the powers and get: (42)3 = 46 = 4096. Notice that this is the same answer as problem 2c.
3 a) x5 · x9 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: x5 · x9 = x5+9 = x14.
b) y3 · y4 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: y3 · y4 = y3+4 = y7.
c) x · x5 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: x · x5 = x1+5 = x6.
d) x3 · x5 · x10 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: x3 · x5 · x10 = x3+5+10 = x18.
4 a) a3b · b4b =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: a3b · a4b = a3b+4b = a7b.
b) ya+1 · ya-1 =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: ya+1 · ya-1 = ya+1+a-1 = y2a.
c) z1+a · z1-a =
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: z1+a · z1-a = = z1+a+1-a = z2.
d) x6 · x-6
Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: x6 · x-6 = x6+(-6) = x0 = 1.
5 a) x8/x3 =
Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: x8/x3 = x8-3 = x5.
b) x5/x-5 =
Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: x5/x-5 = x5-(-5) = x10.
c) y3/y5 =
Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: y3/y5 = y3-5 = y-2. We can use Rule Number Five to rewrite the answer as y-2 = 1/y2, if we wish. You will find that sometimes you prefer the answer in one form, and sometimes in the other form. Both are correct.
d) z2/z-4 =
Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: z2/z-4 = z2-(-4) = z6.
6 a) (x2)7 =
Rule Number Four says that when a number raised to a power is raised to a power, we multiply the powers. Here we get: (x2)7 = x2·7 = x14.
b) (y4)-2 =
Rule Number Four says that when a number raised to a power is raised to a power, we multiply the powers. Here we get: (y4)-2 = y4·(-2) = y-8. We can use Rule Number Five to rewrite the answer as y-8 = 1/y8, if we wish. You will find that sometimes you prefer the answer in one form; sometimes in the other. Both are correct.
c) (z-5)0 =
Any number raised to the power 0 (except 0 itself) is 1, so (z-5)0 = 1.
d) (a8)1/2 =
Rule Number Four says that when a number raised to a power is raised to a power, we multiply the powers. Here we get: (a8)1/2 = a8·1/2 = a4.
7 a) [square root of y] =
Rule Number Six says that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so here we get: [square root of y] = y1/2.
(Continues...)
Excerpted from Attacking Problems in Logarithms and Exponential Functions by David S. Kahn. Copyright © 2015 David S. Kahn. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
