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## Overview

This original volume offers a concise, highly focused review of what high school and beginning college students need to know in order to solve problems in logarithms and exponential functions. Numerous rigorously tested examples and coherent to-the-point explanations, presented in an easy-to-follow format, provide valuable tools for conquering this challenging subject.

The treatment is organized in a way that permits readers to advance sequentially or skip around between chapters. An essential companion volume to the author's *Attacking Trigonometry Problems, *this book will equip students with the skills they will need to successfully approach the problems in logarithms and exponential functions that they will encounter on exams.

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## Product Details

ISBN-13: | 9780486793467 |
---|---|

Publisher: | Dover Publications |

Publication date: | 10/21/2015 |

Series: | Dover Books on Mathematics Series |

Pages: | 112 |

Sales rank: | 398,666 |

Product dimensions: | 5.30(w) x 8.30(h) x 0.40(d) |

## About the Author

David S. Kahn is a Visiting Assistant Professor of Mathematics at the Center for Science and Math Education at Stony Brook University. For more than 20 years, he has taught mathematics to students at a variety of levels. He is the author of *Cracking the AP Calculus AB & BC Exams,* and his other Dover book is *Attacking Trigonometry Problems.*

## 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

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 *x*1, we multiply *x*, one time, so *x*1 = *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, *x*3 and *x*5. What happens when we multiply them together? Remember that *x*3 = *x·x·x* and *x*5 = *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 *x*3 · *x*5 = *x*8. 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

*x*5/

*x*3 =

*x*5-3 =

*x*2. 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 *x*5 and *x*5. 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 *x*5/*x*5 = *x*5-5 = *x*0 = 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 *x*0 = 1!

What does it mean to raise an exponent to an exponent? That is, what does it mean if we have an expression like (*x*2)3? This means that we have (*x*2)3 = *x*2 · *x*2 · *x*2. Well, using our rule for multiplying exponential expressions, we add the exponents and get (*x*2)3 = *x*2 · *x*2 · *x*2 = *x*6. 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 *x*0-4. From our rule above, we know that this means *x*-4 = *x*0-4 = *x*0/*x*4. But we also know that *x*0 = 1, so *x*-4 = *x*0-4 = *x*0/*x*4 = 1/*x*4.

Let's do this again, but with *xa*. We get *x-a* = *x*0-a = *x*0/*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 *x*1/*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 (*x*1/*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 *x*0 = 1 and *x*1 = *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) *x*5 · *x*9 = b) *y*3 · *y*4 = c) *x* · *x*5 = d) *x*3 · *x*5 · *x*10 =

4 a) a3b · a4b = b) ya+1 · ya-1 = c) z1+a · z1-a = d) x6 · x-6 =

5 a) *x*8/*x*3 = b) *x*5/*x*-5 = c) *y*3/*y*5 = d) *z*2/*z*-4 =

6 a) (*x*2)1 = b) (*y*4)-2 = c) (*z*-5)0 = d) (*a*8)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) *x*5 · *x*9 =

Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: *x*5 · *x*9 = *x*5+9 = *x*14.

b) *y*3 · *y*4 =

Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: *y*3 · *y*4 = *y*3+4 = y7.

c) *x · x*5 =

Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: *x · x*5 = *x*1+5 = *x*6.

d) *x*3 · *x*5 · *x*10 =

Rule Number Two says that when we multiply two numbers with the same bases, we add the exponents. Here we get: *x*3 · *x*5 · *x*10 = *x*3+5+10 = *x*18.

4 a) *a*3*b* · *b*4*b* =

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) *x*6 · *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) *x*8/*x*3 =

Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: *x*8/*x*3 = *x*8-3 = *x*5.

b) *x*5/*x*-5 =

Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: *x*5/*x*-5 = *x*5-(-5) = *x*10.

c) *y*3/*y*5 =

Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: *y*3/*y*5 = *y*3-5 = *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) *z*2/*z*-4 =

Rule Number Three says that when we divide two numbers with the same bases, we subtract the exponents. Here we get: *z*2/*z*-4 = *z*2-(-4) = *z*6.

6 a) (*x*2)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: (*x*2)7 = *x*2·7 = *x*14.

b) (*y*4)-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: (*y*4)-2 = *y*4·(-2) = *y*-8. We can use Rule Number Five to rewrite the answer as *y*-8 = 1/*y*8, 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) (*a*8)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: (*a*8)1/2 = *a*8·1/2 = *a*4.

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*] = *y*1/2.

*(Continues...)*

Excerpted fromAttacking Problems in Logarithms and Exponential FunctionsbyDavid S. Kahn. Copyright © 2015 David S. Kahn. Excerpted by permission of Dover Publications, Inc..

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## Table of Contents

#### Contents

Unit One Seven Simple Rules for Working with Exponents, 1,Unit Two Exponential Expressions, 9,

Unit Three Scientific Notation, 18,

Unit Four Graphs of Exponential Functions, 26,

Unit Five Logarithms, 39,

Unit Six Log Laws, 45,

Unit Seven Exponential Growth and Natural Logarithms, 52,

Unit Eight Graphs of Logarithmic Functions, 63,

Unit Nine Problems That Use Exponentials, 77,

Unit Ten Problems That Use Logarithms, 88,