Let's make sure we understand
this logarithm notation.
So I want to find logarithms
for the same number,
for 64, but with two
different bases, 2 and 4.
So this is a really
important exercise.
It's because the
bases in logarithms
can cause a lot of confusion.
So what are we looking for?
Let's say log to
the base of 2 of 64.
Well, we're trying
to work out how
to express 64 as a power of 2.
So for the first
guy, what do we need?
We need a number, let's say
a, such that 2 to the power a
is equal to 64.
Now, this guy is a
different story, right?
We need a number, let's
say d, such that 4 to the d
is equal to 64.
That's what we're looking for.
it's two different things.
So in the first case,
if we find the a,
that gives us log
to the base 2 of 64.
And in the second case, we have
d, would be log to the base
4 of 64.
So it's important to understand
that the base is very, very
important in the
story of a logarithm.
So an observation--
very simple, really.
I mean, 64 is a power
of 2, thank goodness.
And 2 to the power 6 is equal
to 2 cubed multiplied by itself.
It's 8 times 8.
That's equal to 64.
So this is [? what? ?]
This is 64.
And 4 to the 3--
same kind of story--
is equal to 64.
So that is saying, 2
to the 6 and 4 to the 3
are exactly the same number.
They're both 64.
But because the
bases are different,
they kind of look different.
So putting this
together tells me what?
Log to the base 2
of 64 is equal to 6,
and log to the base 4
of 64 is equal to 3.
So this is an important
exercise in understanding
what the notation means,
and also understanding
that the base really
matters when you're
doing a computation,
and your job is
to find some exponent, which
does the appropriate thing
for the given base.
Log to the base
2, the base is 2.
Log to the base
4, the base is 4.
