www.mathcentre.ac.uk/.../Logarithms.mp4

In this video, we're going to be
looking at logarithms, but

before we can deal with
logarithms, what we need to have

a look at our indices 16.

We know that we can write 16 as
a power of two, so this is 2.

Raised to the power 4.

Now need to give some names to
some of these numbers.

So this number here 4.

Gets called a number of
different things. I just called

it a power.

Sometimes it gets called.

Exponent

Sometimes. Index.
So that's the four. What about

this number? This number doesn't
usually get a name, but it does

have one. It's referred to
as the base.

So another example might be

64. 64 is 8 to
the power 2.

So in this case two is
the power, the exponent or

the index, and here 8 is
the base.

So having got the language clear
for indices or powers, what

about logarithms? Well.

Let's see what the motivation
might be for wanting to use

logarithms.

No.
As we've just

seen, 16 is 2
raised to the

power of 4.

8. Similarly, is 2 raised
to the power of 3?

Now, supposing we want it to
multiply 16 by 8, well, one way

of doing it would be long

multiplication. We can do that
and the answer is 128.

That is another way rather than
having to go through this

multiplication. Some which if
the numbers were other than 16

and eight would be very long.
Can we not make use of the way

we've expressed these?

16 times by 8.

Is 2 to the power four times by
2 to the power 3.

From laws of indices, we know
that when we do this kind of

calculation, what we do is we
simply add the indices together.

So what was a
multiplication? Some we've

reduced to an addition
some.

Similarly, we could do this
sort of thing with division,

so if we have 16 / 8, that
would be 2 to the power, 4 / 2

to the power three, and that
would just be too, because we

would subtract the indices.

Now, if we had a table of

numbers. Where we listed these
powers then all we would need to

do is look them up.

Do addition. And then look back
again at what this 2 to the

power 7 actually meant that it

means 128. Now this idea
is the whole basis of

logarithms. And they were
devised in the late 16th

century by two mathematicians
working in dependently John

Napier and Henry Briggs.

So what exactly is

a logarithm? Let's
start with this 16

is 2 to the power 4.

This is the power index or the
exponent, and this is the base.

If we take the logarithm, which
we usually write as log to

base 2. Of this number
16, then the logarithm is

4. This power index
or exponent becomes

the logarithm, so we
take logs to a base.

And it's the base that gets
raised to a particular power in

indices, and that power becomes
the logarithm. So these two are

equivalent statements. If we
write one, we automatically

imply the other.

So if I say that 64 is
8 to the power, two, that is

exactly the same as saying that
the log of 64.

Two base eight is 2. These two
statements are exactly the same.

So if I write a statement down
this side, is the exactly the

same statement written down
here? They both mean the same

thing, so this side I say the
log to base three of 27 is 3.

what I am saying is that 3 to
the power three is equal to 27.

So these statements down the
left here or exactly the same as

these statements down the right.

So. We can write
this down as a general

statement. If we
have a number X.

And we can write X as a

number A. Raised to the power
and then the equivalent

statement in logarithms is to
say that the log.

To the base a of
X is equal to N.

These two are equivalent

statements. Let's just develop a
little bit of that, supposing X

were equal to 10.

Then we can write 10 as 10
to the power one.

So if I now take the
log to base 9:50.

Then be'cause 10 can be written
as 10 to the power one.

Then it's logarithm to the base.
10 must be one. Similarly, if I

had, X is equal to 2.

Then two is just 2
to the power one.

And so if I take the log.

To base two off to again,
that's just one.

So let's make that general. If X
were equal to a, then we know

that we can write that as A to
the power one.

And so that the log to base
a of a is there for one.

And this gives us a general rule
that works for any of our base

numbers a. Let's see if we can
generate some laws of

logarithms. We know that there
are laws of indices. Can we have

similar laws of logarithms, and
in one respect we've already had

the first law?

So. Let's have a look. Let's
take 2 numbers X&Y and what we

want to be able to do is
multiply X&Y together. As a

general thing, so we say X is A
to the power N.

And why is A to the power

N? Now this statement X is A
to the power, N is the

equivalent of saying the log of
X to base a is equal to N.

This statement is the equivalent
of saying the log of Y also to

base a is equal to M.

We want to multiply these two
together so X times by Y is

A to the power N times by
A to the power M.

And what do we get here? A to
the power N times by 8 to the

power N means that we add the
indices together. So that's A to

the N Plus M.

So what is the logarithm?

Two base a of XY.

While quite clearly from the
definition we've had.

X times by Y is A to the power
N plus M, so the logarithm is N

plus N. But N we know is
the log of X to base a log

of X to base A and similarly M
is the log of Y to base A.

And so there we
have our first law

of logarithms. That if we
want to multiply 2 numbers

together, we get the log of the
product by adding the logs of

the two individual numbers.

That's our first law. What about
our second law again?

Let's start with X can be
written as A to the power N.

Now, what if we take X?

And we raise it to the power M.

That would be.

A to the N all raised to
the power M.

And by our laws of indices, we
know that in order to do that,

we multiply the N and the M
together, giving us NN as the

power of A.

So what's the logarithm here?
Log of X to the power M to

the base A.

Equals well, by definition, that
must be this number here. End

times by N.

An times by N.

Equals.
Well, this statement here it's

equivalent statement. Is that
the log of X?

To base a is equal to N, so
instead of NI can write this.

And I must multiply it by NM
times the log of X to base a.

There we have our second law of
logarithms that if we want to

raise a number to a given power.

Then we can do that by taking
the log, multiplying that log by

M. That will give us the log of
the number to the given power,

and then we can look it up in
reverse. So our second law of

logarithms. Our third law, well,
in a way we've already met with

third law because now we've done
multiplication of two different

numbers. Repeated multiplication
of the same number. So what

we're going to look at now is

division. So again, will define
X to be A to the power N.

And why to be A to the power M?

Will write down what that means
in terms of logarithms. Log of X

to the base A is N.

The log of Y to the
base A is N.

And we're going to have a look
at X divided by Y.

So that's A to the N divided by
A to the N and our laws of

indices tell us that we do this
calculation by subtracting these

indices, so that's A to the N

minus M. So what's the
log of this quantity log to

base a of X divided by
Y equals or by definition it

must be this index here.

An minus M.

And N. Is this quantity the log
to base a of X?

Log to base A 4X.

And this quantity M is up here.

The log. Of Y
two base A and. So there

is our third law of logarithms.

That if we want the
log of a quotient.

The log of a division some. Then
we subtract the logs of the two

numbers that were working with.

1 final

point. Final general point.

What happens if we have A
to the power 0?

We know that anything
raised to the power zero

from indices is one.

Well, what does that mean?

About the log.

To base A of

one. Well, since we can write
1 as A to the power zero, that

means that the log of one to
base a must be 0.

And so the log of one in any
base is 0.

Now that does make sense. Let's
just think about it. If you were

doing multiplication. One times
by 6 is 6 * 1 doesn't affect the

six, so if we were to do the
same in logs, we want the log of

one plus the log of six.
Wouldn't want to change the log

of six simply because we knew we
were multiplying by one, so it

makes sense for the log of one
to be 0.

Let's have a look now at some
more examples of calculating

logarithms or various numbers to
particular basis. So let's take

the log of 512 to base 2.

Well, this is the same as asking
what's 512 as a power of two

512 equals 2 to what power?
What's that power up there?

Well, 512 is in fact 2 to
the power 9.

And so by definition.

The log of 512 to base 2.

Is 9.

What about the log?

To base eight of
one over 64.

This is the same as asking
what is one over 64 as

a power. Of
eight.

Well, what is that? We know that
one over 64.

Is 64.

To the minus one.

We also know that 64 is 8
squared, 8 times by 8 and so

that tells us that one over 64
is 8 tool, the minus 2.

And so the logarithm of one over
64 to base eight is minus.

2.

What about? The log

of. 25
Tool base 5.

Well, this is the same as
asking what is 25 written as

5 to the power.

What's that power that index?
What goes there?

Well, what we do know is that
it's 5 squared and so the log of

25 two base five is just two.

I want to do another calculation
which is going to look very

similar to this one and I'll
need to compare it with this

calculation, so I'm going to
repeat this statement over the

page. So we've got log.

Of 25 to base
five, we know that

that's two. Now what if
I interchange the number and the

base? So that I'm asking
the question, what's the log of

five to base 25?

Now that's the same as asking.

If I have 5.

How can I write it as a
power of 25 Watts that

index there?

Well, one of the things I do
know is that the square root of

25 is 5 and I can write a
square root as a power.

1/2

So what that tells me is
that the log of five to

base 25 is 1/2.

And what seems to have happened
here is that by interchanging

the number with the base.

We got one over the logarithm.

Let's check that by looking
at another example.

We know that eight can be
written as 2 to the power

three, and of course that
means that the log of

eight to base two is 3.

But what if we interchange the
number with the base? So we ask

ourselves what's the log of two
to the base 8?

And that's the equivalent of
saying to ourselves. How can I

write 2 as a power of eight?
What goes there? What's that

index? Well, two is
the cube root of

8. Which we write like that. But
another way of writing the cube

root is to write it as 8 to the

power 1/3. That tells us that
the log of two to base eight

is 1/3. And so we see again,
that by interchanging the number

and the base.

We get one over the original

logarithm. And that's true in
general. I haven't proved it by

showing you those two examples,
but I have been able to

demonstrate it and that is true
in general that if we have the

log of B to base A.

Then that is equal
to one over the log

of A to base be.

Now. We've done a lot of work

with different bases. So the
question that we might ask is,

are there any standard basis?
Are logarhythms calculated using

particular basis? And of course
the answer is yes.

What are those bases? Well, one
of the common. The two common

ones is 10. If you look on your
Calculator, you will see a

button that's labeled log.

Just log. That button that's
labeled log gives you logs to

base 10. So for instance, if you
put in 100.

And press the log key. It will
give you the answer to because

the log of 100 to base 10.

That's the log of 10
squared to base. 10

clearly gives us 2.

What's the other common base?
Well, the other common bases

base E. To letter, not a
number you might say, but

remember pie is also a letter,
a Greek letter, and the number

and E and π share something in
common. They both have

infinite decimal expansions,
so E is a very special number.

How big is it? Well to three
decimal places? It's 2.718, but

that's the three decimal places.
Remember, it's got an infinite

decimal expansion. So if you
look on your Calculator, you

will see a button that's got Ln

on it. And that button Ln stands
for logs to base E. So if you

put the number 3 in it and press
Ln, it will give you the log to

base E of three.

These logs have a name.

Sometimes they're called Napier
Ian Logarhythms After John

Napier, and sometimes they're
called natural logarithms.

Logs to base E of the logs that
get used in calculus. So it is

important to know about logs
because they're going to come up

as a regular part of the
calculus, and in particular when

we're solving differential

equations. So let's see if we
can develop a way of using

logarithms and see a use for

them. This isn't going to be a
using terms of calculus, but in

terms of solving a particular

kind of equation. So supposing
we've got 3 to the power

X equals 5.

How would we solve that
equation? The unknown is up

here, it's an index, it's
an exponent.

We need to get it down out
of being an exponent down to

being an ordinary number and
one of the ways of doing

that is to take logarithms,
so I'm going to take the log

of both sides to base 10.

So I'm going to have the log.

Of three to the power, X is
equal to the log of five. Now

notice I haven't written the 10
in on the base.

And that's because I'm using the
convention that I've just

expressed that log now means log
to base 10.

Well, I've got the log of three
raised to the power X and of

course one of the things that I
do know from my laws of logs is

that I'm raising fun raising to
the power X. Then what I need to

do is multiply the logarithm by

X. Well now log of three
and log of five aren't just

numbers, nothing more.

So in the same way that I
might solve an equation such as

three X equals 12 by dividing
both sides by three.

Then I'm going to do exactly the
same with this, and I'm going to

divide both sides by the log of

three. And that calculation can
be finished off using a

Calculator. We take the log of 5
divided by the log of three

using our Calculator and get the
answer for X.

Let's just have a look at one

more example. If we have three
to the X is equal to 5 to the

X minus two. What then again,
the unknown is X and again it's

upstairs, so to speak. It's in
the index in the power and we

need to bring it back downstairs
to the normal level. So again

will take logs of both sides,
log of three to the power X is

equal to the log of 5 to the

power. Minus two in the same
way, if we're raising to the

power, then in order to get the
logarithm we need to multiply.

The log of three by X
and the log of five by

X minus 2.

Well, this is now just an
ordinary equation log of three

and log of five and nothing more
than numbers. So let's multiply

out this bracket this side.

So we're going to have
X log 5 - 2

log 5. Let's gather together
some terms in X and move this as

a number over to this site so
it's positive. So we're going to

add this to both sides, which
will give Me 2 log 5.

And I'm going to take this away
from both sides, so X log 5

minus X log 3.

I've got all my ex is together
here, so I'm going to take X out

as a common factor 2.

Log 5 is X times log 5 minus
log 3 and close the bracket. So

now to get X all I need to
do is to divide both sides by

this number here log 5 minus log
3 because that's all it is. It's

just a number, so let's do that
right. The line down again to

log 5 is.

Equal to X times log
5 minus log 3.

And I'm going to divide both
sides by this number 2 log

5 divided by log 5 minus
log of three is equal to

X. OK, I've got X. You can
put this lump of numbers here

into a Calculator and work them
out. One way of making that

calculation alittle bit simpler
though, is to notice that here

we are subtracting two logs and
if we're subtracting two logs,

that means we are in fact
dividing five by three in terms

of the numbers themselves. So we
can write this as two log 5.

Over log of
5 / 3.

And that gives us perhaps
a slightly better form to

do the calculations in.

Now there's one further thing
that I just want to have a look

at and this will become quite
important when doing calculus

so. Let me take the number

8. And I'm going to raise
2 to the power 8.

Let me

know. Take
logs to base two

of two to the

power 8. Well, I
know what that is, that's eight.

I just think what's happened
there. We started with a number.

We took a base and we raise the
base to that power.

We then took logs to that bass.

Of our resulting answer and we
ended up with the number that we

first started with.

So any fact what we did here
undid what we've done there.

Now when that happens, what
we're saying is we have got

inverse operations.

That what this operation
does, raising the base to the

given power is undone by what
this operation does.

Does it work the
other way around?

So let's start with eight again.

This time, let's take the

log. To base.

Two of eight.

Well, I know the
log.

To base two of eight, well, I
need to write the 18 a different

way I need it is 2 to the power
three and then of course gives

me the answer straight away.
This number is 3.

Now I take my base
#2 and I raise it

to this power 3.

And the answers 8.

So again, I've done these two
operations in a different order.

But the number that I started
with and the number that I end

up with are exactly the same.

So what I've got here are two
inverse operations. What one

does the other one undoes? So in
terms of our standard basis,

what does that mean? It means
that the natural log of E to

the X. Here I've taken X as
a power, and I've raised the

base E to that power X and then
I've taken the log.

The log undoes what I did
with the X and leaves me

with X again. Similarly, if I

Hav E. Raised to the power,
the log of X, this is again X

because I've used these two
process is these two things.

Again in combination there
inverse operations, so they must

undo what the other one has

done. Similarly.
If we take our base 10,

then the log.

Of 10 Raised to
the power, X is just X.

And if we take.

10 And we raise it to
the power. The log of X. Then

again, the answer is just X
because we have undone.

By raising 10 to this power, we
have undone what we did by

taking the log of X to base 10.

This is the particularly
important one when you're doing

calculus. This is the one that
you will need to bear in mind.