Finding logarithms is
difficult. And for most numbers
and most bases, you're not
going to get an exact value.
But under certain circumstances,
it's possible to do it.
And let's take a think
about these two questions.
So log to the base 4 of
2-- so what do we need?
Let's just wrap our
heads around this.
We need a number, let's
say a, such that 4 to the a
is equal to 2.
That's what we're
looking for here.
This is the meaning
of the logarithm.
What do I know?
Well, 4 to the a--
4 is bigger than
2, so it couldn't
be 1, 2, 3, anything like that.
It's going to have
to be less than 1.
What do I know?
Well, I know the square root
of 4 is equal to 2, isn't it?
The positive square
root of 4 is equal to 2.
So the fourth power
of 1/2 is equal to.
2.
So, observation-- the square
root of 4 is equal to-- well,
it's 4 to the power
of 1/2, as I said.
By definition,
that's equal to 2.
So this tells us that log to
the base 4 of 2 is equal to 1/2.
Now, b-- What do we need?
Well, same thing as before.
We're looking for some number--
I'll call it d again.
So we need a d such that 4 to
the power d is equal to 32.
And this seems quite a
little more difficult.
We saw on the previous
comprehension check
that if it was 64 instead
of 32, it was good,
because I had 4 to the 3.
But it's a bit of a
problem, because if I do 4
to the power 2, it's equal
to 16, which is less than 32.
If I do 4 to the 3, it's 64.
So it jumps over.
So intuitively, I think my d
has to be somewhere between 2
and 3.
But what exactly is it?
So what I'm going to do is
something really clever.
You can see several
ways to do this,
but this is a useful
way to think about it.
If I have the
following-- if I have 2--
so I know the following is true.
So 4-- this is a tricky example.
4 to the power of
1/2 is equal to 2.
Now what about 32
as a power of 2?
What's that equal to?
Well, 2 to the power 4 is 16.
2 to the power 5 is equal to 32.
So why don't I take
this expression
and raise everything
to the power 5?
So what's that equal to?
That's equal to 32.
But notice, because I've
expressed to 2 a power of 4,
it's 4 to the power
of 1/2, I can now
use one of my laws of exponents.
So this would now be 4,
5/2, over 2 is equal to 32.
So I've done something
really clever here.
That's why it was a
part a and a part b.
So finally, what
does this tell you?
It tells you log to the base
4 of 32 is equal to 5/2.
That's a difficult example.
You've really got
to think carefully
about the meaning
of the logarithm
and how you could possibly
use it to get anywhere.
And the real thing is--
I mean, it's recognizing that 2
to the power 5 is equal to 32.
When you spot that, and
you really follow through
from part a, you end up
being able to express 32
as a power of 4, which gives
you log to the base 4 of 32.
So a difficult example,
but a really good one.
