 
In this screencast we will be looking at
what logarithms
represent (how they're defined), how we can work out
the numerical values that logarithms
take - or at least
estimate those values - and then we'll
look at one of the main applications
that
logarithms are used for in practice.
Now, you're probably familiar with what
we call a power statement:
2 to the power 3 equals 8.
What we say here, in general, is that we
have a base,
in this case 2, which is raised to the
power of 3
and what that produces is the number 8,
because 2 times 2 times 2
produces 8.
Logarithms are the inverse,
or the opposite statement, to the one
we see here.
When we ask for the "logarithm base 2 of 8"
we are asking
"what power of the base that's given in
the logarithm (in this case 2)
produces the number 8" and the answer,
as we've seen above, of course,
is that 2 to the power of 3 equals 8,
so the answer to our logarithm is the
numerical value of 3.
Let's try that out on a couple of
examples because that will tell us
how logarithms are worked out in
practice when you
meet them in your work. Let's look at
this one:
"logarithm base 3 of nine".
What does that equal? Well, what we're
asking here
is "what power of the base 3
produces 9?" Have a think about that for a
moment and
pause the video if you need some time.
Did you get the answer 2?
That's because 3 multiplied by 3
produces 9, so logarithm base 3 of 9
is 2. Let's look at another one.
Here we have "logarithm base 2 of 16".
What does that equal? 
Have a think about that for a moment.
What you should have done
in your head is think "what power of 2
produces 16?"
In other words, "how many times do I have
to multiply 2 by itself to get 16?"
If you think about that
for a moment you should find the answer
is 4.
One more.
Have a think about that one for a
moment.
In this case you should have asked
yourself
"what power of 10 produces 100?"
That one shouldn't be too hard because
we live in a base 10
number system. Our number system has 10
digits in it
going from 0 to 9 so powers of 10
are quite easy to work out and it doesn't
take much to realize that
10 times 10 is 100, therefore the
answer
to that logarithm must be 2. Now those
preceding examples were fairly
straightforward. It's quite easy to work
out
what power of the base produces the
number
being subjected to the logarithm. What if
we had a situation like this:
Here we are after the log base 10 of the number 550.
Now this one is going to be much harder
to work out.
Have a think about what the answer might be.
See if you can come up with a number
that's going to be
roughly in the ballpark of that
logarithm.
One way to tackle this problem
is to consider known logarithms that are
close
to 550. So, for example
logarithm base 10 of 100 we saw
previously was equal to 2 and 100 is a
number that's a bit smaller than 550.
On the other side of 550 we might
recognize that
logarithm base 10 of 1000 is 3,
so it would seem logical that the answer
to the logarithm in the middle
should be somewhere between 2 and
3. It's possible that you might also
pick up that
550 is exactly halfway between
100 and 1000, so we might suspect
that the logarithm of that number is
going to be roundabout
halfway between 2 and 3. Now beyond
those
speculations we can't say exactly what
that
logarithm is going to be. In order to find
out
the logarithm base 10 of 550 we're
going to need
one of these - a calculator. Calculators have
two kinds of logarithm buttons available
(usually).
Your typical scientific calculator like
this one here
will probably have a "log" button which
is the one you see here.
Now to save space it just writes "log" and
it's assumed
this is a logarithm with a base of 10. If
you input the number 550
then hit the equal sign it will tell you
the number you see in the display here.
So that confirms
a couple things. It confirms that the
logarithm
base 10 of 550 is between 2
and 3. It's also roundabout halfway
between 2 and 3 but not exactly
halfway.
It's also what we call an "infinite
decimal",
or, if you know something about the
different number systems, it's what's called
an "irrational number".
It has a decimal expansion that goes on
for ever
as indicated by those three dots on the
screen
without ever repeating itself.
And that's why you need a calculator
to work out most logarithms.
But there's no reason why you can't look
at a logarithm, like the one we've been
working with here,
and get a rough idea of the ballpark in
which it should be.
If you know something about the
numerical value of a logarithm
then you're much closer to being able to
use them effectively
and understand what they're doing for you.
To finish this screencast let's look at
what logarithms can do to help us in our
mathematical and scientific work. Here's a
typical
problem that we might encounter in
Biology or
in Business and Economics. A population
is growing at a rate proportional to
its size at any one time. So here we have
a population
that increases by a factor of 10, or
grows tenfold,
every year. And what we often do
is produce a visual representation of that.
We start with
some years going along a horizontal axis,
represented by the letter t there.
Year 1, 2, 3 and 4, and we use a
vertical axis
to plot the size of the population at
various times.
So let's say we started out with
just one individual in the population so
we'll plot them there on the vertical axis
at 1. We know
that 1 year later there will be 10
individuals in that population.
After 2 years the population will have
grown ten-fold again,
which means there will be 100 individuals
in that population.
Now we can't plot that on the vertical
axis we've used
because it finishes at 10, so what I'll
do is I'll
collapse that 10 down much lower
and expand the axis to include a
100, so I can
now plot the population after two years.
Again, a year later, the population will have
increased
by factor of 10 again and there will be
1000 members,
so I'll drop that 100 down and make
the axis
even longer, going up to 1000 and include
those people there. And as you can see
it's
a reasonably difficult thing to plot a
graph of.
The early population sizes at time 0
1 and 2 are starting to cluster in a
sort of difficult to view clump at
the bottom
of the axis. Let's go one step further,
after 4 years our population has a
whopping
10,000 members. If we join those dots
with a smooth curve,
indicating continuous growth of that
population,
we get a picture that looks like this.
Classic exponential growth. The population starts out
relatively small and then at some point
it seems to explode
into a much more rapidly growing
population.
This is at the heart of a lot of
biological systems, as you can imagine,
and also has a lot to do with compound
interest
in applications in Economics and
Business. Now, as we saw earlier,
those figures on the vertical axis
are all powers of 10.1000 is 10 raised to the power 3
and at the top end of our new scale, 10 to the power of 4 is 10,000.
So what we can actually do
is, rather than using what's called a
"linear" scale for that vertical axis,
we just plot the powers 10
instead. So if we do that,
we get this picture. So the 4 at the
top that vertical axis is representing
10,000,
the 3 is representing 1000, the
2 is representing 100
and so on. And what you see now is that
that exponential growth
has been changed to a straight line
model.
Now, straight line models are much easier
to deal with
than exponential or any other type of
curved model,
so using this approach is going to
make
the representation of the information a
lot easier. The gap between each number
on that vertical scale represents a
multiple
of 10. So it's what we call a "non-linear" scale.
But each of those numbers still represents
a power of 10. In fact 4
is the same as logarithm base 10 of 10,000.
3 is the logarithm base 10 of 1000.
We saw earlier log base 10 of 100 is 2
and similarly log base 10 of 10 turns
out to be 1.
So we refer to this kind of picture as a
"log-linear" plot. The vertical axis represents
the logarithm of
the population size
and the horizontal axis is an ordinary linear scale
(like a ruler).
That's a very simple example of
population growth,
an application of logarithms to
the graphing of important information.
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