Introduction to Logarithms

An introduction to logarithms

Welcome to the logarithm presentation.

Let me write down the word "logarithm" just because

it is another strange an unusual word, like "hypotenuse",

and it's good to at least see it once.

Let me get the pen tool working.

Logarithm.

This is one of my most misspelled words.

I went to MIT and actually one of the a cappella groups there,

they were called the Logarhythms.

Like rhythm, like music.

But anyway, I'm digressing.

So what is a logarithm?

Well, the easiest way to explain what a logarithm is is

to have first-- I guess it's just to say it's the inverse of

taking the exponent of something.

Let me explain.

If I said that two to the third power-- well, we know that

from the exponent modules.

two the third power, well that's equal to eight.

And once again, this is a two, it's not a z.

two to the third power is eight, so it actually turns out that

log-- and log is short for the word logarithm.

Log base two of eight is equal to three.

I think when you look at that you're trying to say oh,

that's trying to make a little bit of sense.

What this says, if I were to ask you what log base two of

eight is, this says two to the what power is equal to eight?

So the answer to a logarithm-- you can say the answer to this

logarithm expression, or if you evaluate this logarithm

expression, you should get a number that is really the

exponent that you would have to raised two to to get eight.

And once again, that's three.

Let's do a couple more examples and I think you might get it.

If I were to say log-- what happened to my pen?

log base four of sixty-four is equal to x.

Another way of rewriting this exact equation is to say four to

the x power is equal to sixty-four.

Or another way to think about it, four to what

power is equal to sixty-four?

Well, we know that four to the third power is sixty-four.

So we know that in this case, this equals three.

So log base four of sixty-four is equal to three.

Let me do a bunch of more examples and I think the more

examples you see, it'll start to make some sense.

Logarithms are a simple idea, but I think they can get

confusing because they're the inverse of exponentiation,

which is sometimes itself, a confusing concept.

So what is log base ten of let's say, one million.

Put some commas here to make sure.

So this equals question mark.

Well, all we have to ask ourselves is ten to what power

is equal to one million.

And ten to any power is actually equal to one followed by the

power of-- if you say ten of the fifth power, that's equal

to one followed by five zero's.

So if we have one followed by six zero's this is the same thing

as ten to the sixth power.

So ten to the sixth power is equal to one million.

So since ten to the sixth power is equal to one million log base

ten of one million is equal to six.

Just remember, this six is an exponent that we raise ten

to to get the one million.

I know I'm saying this in a hundred different ways and

hopefully, one or two of these million different ways that I'm

explaining it actually will make sense.

Let's do some more.

Actually, I'll do even a slightly confusing one.

log base one / two of one / eight.

Let's say that that equals x.

So let's just remind ourselves, that's just

like saying one / two-- whoops.

one / two.

That's supposed to be parentheses.

To the x power is equal to one / eight.

Well, we know that one / two to the third power is equal to one / eight.

So log base one / two of one / eight is equal to three.

Let me do a bunch of more problems.

Actually, let me mix it up a little bit.

Let's say that log base x of twenty-seven is equal to three.

What's x?

Well, just like what we did before, this says that x to the

third power is equal to twenty-seven.

Or x is equal to the cubed root of twenty-seven.

And all that means is that there's some number times

itself three times that equals twenty-seven.

And I think at this point you know that that

number would be three.

x equals three.

So we could write log base three of twenty-seven is equal to three.

Let me think of another example.

I'm only doing relatively small numbers because I don't have

a calculator with me and I have to do them in my head.

So what is log-- let me think about this.

What is log base one hundred of one?

This is a trick problem.

So once again, let's just say that this is equal

to question mark.

So remember this is log base one hundred hundred of one.

So this says one hundred to the question mark power

is equal to one.

Well, what do we have to raise-- if we have any number

and we raise it to what power, when do we get one?

Well, if you remember from the exponent rules, or actually not

the exponent rules, from the exponent modules, anything to

the zero-th power is equal to one.

So we could say one hundred to the zero power equals one.

So we could say log base one hundred hundred of one is equal to zero

because one hundred to the zero-th power is equal to one.

Let me ask another question.

What if I were to ask you log, let's say base two of zero?

So what is that equal to?

Well, what I'm asking you, I'm saying two-- let's

say that equals x.

two to some power x is equal to zero.

So what is x?

Well, is there anything that I can raise two to

the power of to get zero?

No.

So this is undefined.

Undefined or no solution.

There's no number that I can raise two to the

power of and get zero.

Similarly if I were to ask you log base three of

let's say, negative one.

And we're assuming we're dealing with the real numbers,

which are most of the numbers that I think at this point

you have dealt with.

There's nothing I can raise three three to the power of to

get a negative number, so this is undefined.

So as long as you have a positive base here, this

number, in order to be defined, has to be greater than-- well,

it has to be greater than or equal-- no.

It has to be greater than zero.

Not equal to.

It cannot be zero and it cannot be negative.

Let's do a couple more problems.

I think I have another minute and a half.

You're already prepared to do the level one logarithms module,

but let's do a couple of more.

What is log base eight-- I'm going to do a slightly

tricky one-- of one / sixty-four.

Interesting.

We know that log base eight of sixty-four would equal two, right?

Because eight squared is equal to sixty-four.

But eight to what power equals one / sixty-four?

Well, we learned from the negative exponent module that

that is equal to negative two.

If you remember, eight to the negative two power is the same

thing as one / eight to the two power.

eight squared, which is equal to one / sixty-four.

Interesting.

I'll leave this for you to think about.

When you take the inverse of whatever you're taking the

logarithm of, it turns the answer negative.

And we'll do a lot more logarithm problems and explore

a lot more of the properties of logarithms in future modules.

But I think you're ready at this point to do the level one

logarithm set of exercises.

See you in the next module.