This is the first of several videos

on logarithms and their algebraic properties

First, what is a logarithm?

What do we mean by log?

So... say that the logarithm of x equals 
some number, "a"

if 10 raised to the "a" power equals x.

That's the definition of a log.

It's a slightly indirect definition,

but that's the definition that I always 
go back to when thinking about

what logs are.

There's another way to say this.

You can basically write this out in a sentence.

Let me do that.

Another way of saying this as a sentence is

that log x is that number which,

if you put in an exponent above 10,

gives x.

In other words, 10 to the log x, is x.

This tells us that logarithming, taking a log

and then exponentiating, they undo each other

They are actually inverses of each other.

One does something the other 
undoes something.

Logs undo exponentiation.

If you log something and then exponentiate it

you're right back to where you started.

This is the basic equation.

Really these two say the same thing.

That we'll use when we start 
to think about logs

Let's do a few examples using
this definition.

Let's see... log of 100.

What would that be equal to?

I claim that it's equal to 2.

Why?

Because 10 to the 2 equals 100.

2 is that number which, if I put it up
in an exponent for 10

gives me 100. So, log of 100 is 2.

We could do another example.

Log of 100,000... what does that equal?

I claim that equals 5.

Why?

Because 10 to the 5 is 100,000.

OK. Let's do one more example.

Maybe I'll just write this again.

That's the thing we remember for logs.

So what about

log of 500?

I'm looking for some number which, if I

put it in the exponent, gives me 500.

Here, without a calculator, I can't
do that exactly.

But I can figure out some stuff... so

10 to the 2 is 100.

10 to the 3 is 1000.

so 500, because it's between
these two numbers...

the log of 500 must be between 2 and 3.

Let me write that:
log of 500 is between 2 and 3.

If you needed an exact value,

you could look it up on a calculator.

So, I can do that. 500... log...

About 2.699

It turns out that log of 500 is about 2.699

Why?

Because 10 to the 2.699 is going to 
be about 500.

I can test that out on a calculator, too.

10 raised to the 2.699... sure enough,
is about 500... 500.03

Alright. This down here, you would need 
a calculator to do.

To figure out an exact value,

or a closer to exact value.

But without a calculator, you should be

able to know that log 500 is between

2 and 3. Because 10 to the 2 is 100

and 10 to the 3 is 1000.

In the next couple of quizzes,

you'll practice thinking about logs

using this basic relationship

and along the way, you'll discover

values for a few special numbers..

logarithms for a few special numbers.

So, give the quizzes a try, and if
any of them are confusing

be sure to check out the solutions.