**Chapter 1**

“Crucy! Get over here and help load the ship.”

“Coming Brogalia,” I said as my long gangly legs bounded towards her.

I couldn’t believe it. Me, on a spaceship with her—the most gorgeous babe on planet Earth, going off on the adventure of a lifetime. It was going to be incredible. No, unforgettable. I was going to be holed up in the Norrobia Spacecraft II for a whole month with Brogalia and 24 others on a mission to colonize the first planet in the Andromeda Galaxy.

“Brroggaliiaa…” I stammered. “Where should I put the CRB #46 plant canisters?”

“Just put them in storage unit 3 down in Sector 5. Do you remember the password?”

“Yes,” I said as I made my way down the steps toward the spaceship.

Brogalia programmed all of the passwords on the spaceship and there are practically hundreds of them. I couldn’t really remember the password (who could?), but I wasn’t going to tell her that.

Ever since we started high school I have been poring over stacks and stacks of math books after school. I was never really any good at math, but I have found an unquenchable desire to learn all I can to impress her, and it appears to be paying off.

Brogalia was one of those child prodigies that seemed to know everything about anything. Even though we just barely turned 18, she got put in charge of the computational aspects of the mission. There were thousands of candidates, but her creative mathematics won the selection committee over.

The only reason I got to be part of this mission was because my father was the lead scientist who pioneered the new colonization protocol system that automatically begins the process of making uninhabitable planets habitable. It’s a highly complex system, and my father was the first one to discover a mathematical equation that describes the relationships between all organisms within an ecosystem including plants, animals, humans, and even bacteria. Because of his breakthrough, the President gave him the privilege of taking his whole family on the expedition.

Finally, I made it to sector 5. Now what was that password? I knew it had something to do with the number 10 because that was Brogalia’s favorite number. She was always so tricky with her passwords. I think I remember her saying that the password was the exponent you raise 10 to get some number. But what was the number? Oh, that’s right. The number is 678 since this is sector 5 and for each sector she chose the three digits in a row after the sector number. Thus for sector 4 she chose the numbers 567 and for 6 she chose 789. But this is sector 5 so the number we need is 678. she wouldn’t make this the password because that would be too easy. The password is the exponent you raise 10 to get 678. Now how can I represent this in a math sentence? How about

That’s the exponent I’m looking for. If I can figure out what is I’ll have the password. I haven’t solved very many problems like this before so I’ll have to play around with it. I know that

which is too little, and

but that’s too much. So isn’t or . I guess the exponent must be a number between 2 and 3. I didn’t even know you could do that. It looks like I’ll have to guess. I’ll start with 2.5. To plug this in my calculator I need to fine the ^ button which means “to the power of.” This raises a number to any exponent desired. When it is written out it looks like , but to type it into my calculator I need to push 10 ^ 2.5 and then the = button. Plugging 10^2.5 = into my calculator gives me . I better write this down:

That’s still too small. I need . How about 10^2.9? This gives me

This is now too high. How about

Still too low. This might take me a while. I wonder if there’s a better way. I know the password is 4 numbers long so it looks like I’ll have to find three decimal places after the number to find the password. It might take me all day guessing like this and Brogalia will be wondering what’s taking me so long. Think. There has to be a faster way.

I remember reading something about John Napier, a 16^{th} and 17^{th} century mathematician, who wanted to make calculating numbers like this one easier. He called his process a *logarithm* because it had something to do with saving time calculating large numbers. A *logarithm* is an exponent. That’s exactly what I am looking for, an exponent. Since a *logarithm *is an operation, I can apply this operation to both sides of an equation. I better work with my original problem I was trying to solve

I need to get that all by itself. The operation appears abbreviated as on a calculator. If I apply the operation or log to both sides of this equation I get

I need to somehow get that on one side of the equation just like I’ve done so many times before when solving equations. But how do I get an down so it’s not an exponent? I remember a law of logarithms (I’ll call it the **exponent rule for logarithms**) that may be helpful

where is a positive number. In this case . This means that

is the same as

which is exactly what I wanted to do. I wanted to bring that exponent down. This makes the equation

I’ve solved lots of equations for so I know that I just need to divide both sides of the equation by since that will give me on the side of the equation with and I know that any fraction where the numerator or top of the fraction is equal to the denominator or bottom of the fraction are equal is just 1. Like when you have or . These are both equal to 1. Using this fact gives me

Now I just need to punch log (678) / log (10) = into my calculator. I’ll write it down.

Alright! I finally figured out the password. The password is only 4 digits long so it looks like it is .

“Password accepted,” said an automated women’s voice as the glass door slid open. I better hurry and get these canisters put away before Brogalia starts wondering why I’ve been gone so long. On second thought, she may send me to a different sector next time so I better hurry and practice solving similar problems so it doesn’t take me so long to figure out other passwords.

But before I crunch out some numbers I better review some things I learned a while back about logarithms.

**Changing from exponential form to logarithm form or the other way around:**

Brogalia likes to mix things up with her passwords. The password usually has the form

but sometimes she makes the password the variable , and sometimes the variable like the one for sector which was written as . For example, sometimes she formulates the password by picking numbers for and and makes the password whatever would be. The variable is called the **base**. So if she picked and , we would change

and write

Then to find the password, solve for . In this case, the password would be since

She hardly ever makes it that easy though. She usually rewrites things like like this

This is an equivalent way of writing because a logarithm is the inverse of an exponential. This is called exponential form

and this is called logarithm form

and these are equivalent statements. This is just one specific case though. It can be written with any number , and . This is **exponential form**

and this is** logarithm form**

and these are also **equivalent **statements**.** I better do some quick examples. Suppose I need to rewrite the exponential

in logarithm form. So the base , , and . To put this in logarithm form I have

I’ll try to do one backwards now. If I have an equation in **logarithm form **and want to convert it** to exponential form** I would do it this way: If I had

then to rewrite it in logarithm form I would write

I think I’m getting the hang of these but I’ll practice a few more in a minute.

I better do one more quick example that’s a little harder before I practice a bunch. Brogalia usually picks more difficult numbers like and and has the password as the unknown variable . In this instance, I need to change

to

and solve for . To solve for , take the log of both sides and use the exponent rule. Remember the **exponent rule** for logarithms is

So my equation becomes

Whenever someone writes or or or log of any number, then we assume the base is . In other words

But if it is written , then the base is 4. So if we had , then we would write since is the base. The log button on a calculator has base 10.

To get by itself in the equation we are trying to solve above, the equation must be converted from its logarithm form back to its exponential from. This means that

can be rewritten in an equivalent form as

since is the base in . To plug this in my calculator I need to type: 10 ^ (log (47)/ 5.678) = and this gives me

I think I’m ready for a bunch of practice.**Chapter 1**

“Crucy! Get over here and help load the ship.”

“Coming Brogalia,” I said as my long gangly legs bounded towards her.

I couldn’t believe it. Me, on a spaceship with her—the most gorgeous babe on planet Earth, going off on the adventure of a lifetime. It was going to be incredible. No, unforgettable. I was going to be holed up in the Norrobia Spacecraft II for a whole month with Brogalia and 24 others on a mission to colonize the first planet in the Andromeda Galaxy.

“Brroggaliiaa…” I stammered. “Where should I put the CRB #46 plant canisters?”

“Just put them in storage unit 3 down in Sector 5. Do you remember the password?”

“Yes,” I said as I made my way down the steps toward the spaceship.

Brogalia programmed all of the passwords on the spaceship and there are practically hundreds of them. I couldn’t really remember the password (who could?), but I wasn’t going to tell her that.

Ever since we started high school I have been poring over stacks and stacks of math books after school. I was never really any good at math, but I have found an unquenchable desire to learn all I can to impress her, and it appears to be paying off.

Brogalia was one of those child prodigies that seemed to know everything about anything. Even though we just barely turned 18, she got put in charge of the computational aspects of the mission. There were thousands of candidates, but her creative mathematics won the selection committee over.

The only reason I got to be part of this mission was because my father was the lead scientist who pioneered the new colonization protocol system that automatically begins the process of making uninhabitable planets habitable. It’s a highly complex system, and my father was the first one to discover a mathematical equation that describes the relationships between all organisms within an ecosystem including plants, animals, humans, and even bacteria. Because of his breakthrough, the President gave him the privilege of taking his whole family on the expedition.

Finally, I made it to sector 5. Now what was that password? I knew it had something to do with the number 10 because that was Brogalia’s favorite number. She was always so tricky with her passwords. I think I remember her saying that the password was the exponent you raise 10 to get some number. But what was the number? Oh, that’s right. The number is 678 since this is sector 5 and for each sector she chose the three digits in a row after the sector number. Thus for sector 4 she chose the numbers 567 and for 6 she chose 789. But this is sector 5 so the number we need is 678. she wouldn’t make this the password because that would be too easy. The password is the exponent you raise 10 to get 678. Now how can I represent this in a math sentence? How about

That’s the exponent I’m looking for. If I can figure out what is I’ll have the password. I haven’t solved very many problems like this before so I’ll have to play around with it. I know that

which is too little, and

but that’s too much. So isn’t or . I guess the exponent must be a number between 2 and 3. I didn’t even know you could do that. It looks like I’ll have to guess. I’ll start with 2.5. To plug this in my calculator I need to fine the ^ button which means “to the power of.” This raises a number to any exponent desired. When it is written out it looks like , but to type it into my calculator I need to push 10 ^ 2.5 and then the = button. Plugging 10^2.5 = into my calculator gives me . I better write this down:

That’s still too small. I need . How about 10^2.9? This gives me

This is now too high. How about

Still too low. This might take me a while. I wonder if there’s a better way. I know the password is 4 numbers long so it looks like I’ll have to find three decimal places after the number to find the password. It might take me all day guessing like this and Brogalia will be wondering what’s taking me so long. Think. There has to be a faster way.

I remember reading something about John Napier, a 16^{th} and 17^{th} century mathematician, who wanted to make calculating numbers like this one easier. He called his process a *logarithm* because it had something to do with saving time calculating large numbers. A *logarithm* is an exponent. That’s exactly what I am looking for, an exponent. Since a *logarithm *is an operation, I can apply this operation to both sides of an equation. I better work with my original problem I was trying to solve

I need to get that all by itself. The operation appears abbreviated as on a calculator. If I apply the operation or log to both sides of this equation I get

I need to somehow get that on one side of the equation just like I’ve done so many times before when solving equations. But how do I get an down so it’s not an exponent? I remember a law of logarithms (I’ll call it the **exponent rule for logarithms**) that may be helpful

where is a positive number. In this case . This means that

is the same as

which is exactly what I wanted to do. I wanted to bring that exponent down. This makes the equation

I’ve solved lots of equations for so I know that I just need to divide both sides of the equation by since that will give me on the side of the equation with and I know that any fraction where the numerator or top of the fraction is equal to the denominator or bottom of the fraction are equal is just 1. Like when you have or . These are both equal to 1. Using this fact gives me

Now I just need to punch log (678) / log (10) = into my calculator. I’ll write it down.

Alright! I finally figured out the password. The password is only 4 digits long so it looks like it is .

“Password accepted,” said an automated women’s voice as the glass door slid open. I better hurry and get these canisters put away before Brogalia starts wondering why I’ve been gone so long. On second thought, she may send me to a different sector next time so I better hurry and practice solving similar problems so it doesn’t take me so long to figure out other passwords.

But before I crunch out some numbers I better review some things I learned a while back about logarithms.

**Changing from exponential form to logarithm form or the other way around:**

Brogalia likes to mix things up with her passwords. The password usually has the form

but sometimes she makes the password the variable , and sometimes the variable like the one for sector which was written as . For example, sometimes she formulates the password by picking numbers for and and makes the password whatever would be. The variable is called the **base**. So if she picked and , we would change

and write

Then to find the password, solve for . In this case, the password would be since

She hardly ever makes it that easy though. She usually rewrites things like like this

This is an equivalent way of writing because a logarithm is the inverse of an exponential. This is called exponential form

and this is called logarithm form

and these are equivalent statements. This is just one specific case though. It can be written with any number , and . This is **exponential form**

and this is** logarithm form**

and these are also **equivalent **statements**.** I better do some quick examples. Suppose I need to rewrite the exponential

in logarithm form. So the base , , and . To put this in logarithm form I have

I’ll try to do one backwards now. If I have an equation in **logarithm form **and want to convert it** to exponential form** I would do it this way: If I had

then to rewrite it in logarithm form I would write

I think I’m getting the hang of these but I’ll practice a few more in a minute.

I better do one more quick example that’s a little harder before I practice a bunch. Brogalia usually picks more difficult numbers like and and has the password as the unknown variable . In this instance, I need to change

to

and solve for . To solve for , take the log of both sides and use the exponent rule. Remember the **exponent rule** for logarithms is

So my equation becomes

Whenever someone writes or or or log of any number, then we assume the base is . In other words

But if it is written , then the base is 4. So if we had , then we would write since is the base. The log button on a calculator has base 10.

To get by itself in the equation we are trying to solve above, the equation must be converted from its logarithm form back to its exponential from. This means that

can be rewritten in an equivalent form as

since is the base in . To plug this in my calculator I need to type: 10 ^ (log (47)/ 5.678) = and this gives me

I think I’m ready for a bunch of practice.