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.