>> In this video we derive
the power property of logs
and use it to do the
following five problems.
This is Part Seven
of logarithms.
We're going to learn the power
property of logs by deriving it.
So let's start with this.
Let's say the log of A base B
equals M. Then that would mean
that the B to the
M equals A. Right?
You take B to the
exponent equals A. So B
to the M equals A. All right,
now here's the question.
What is the log base
B of A to the N. Okay?
So this is a little
bit different.
Well, what I'm going
to do is substitute
in for A what I have right here.
A is actually B to the
M. So I could say well,
this is just really saying the
same thing as log base B of B
to the M raised to
the nth power.
Now from our laws of exponents,
we know we can multiply
the M times M.
So that's the log base
B of B to the M N. Now,
this is another property
we've already gone over.
B to what, well,
equal to the M N,
it's actually just M N.
These bases are the same.
Now although it's written
as M N, I'm just going
to use the commutative
property and write
that as M times N.
You'll see why
on this very last
step I'm going to do.
I'm going to replace M now
with what it was up here,
M could be written as the
log of A base B. So finally,
I end up with N and what
I'm replacing M with,
which is the log of A base B. By
the way, this is hard to prove.
I'm just showing
where this comes from,
as opposed to just all
of a sudden somebody just
tells you what I'm going
to write right now, which
is that the log base B of A
to the N actually equals N
times the log of A base B.
So what you would notice here
is if you have the log base B
of something to the
nth power, you can take
that little exponent
N and it gets moved
out in front of the log.
And there's a reason why this,
hopefully it will
make sense to you.
Remember, logs are
just exponents.
So when you see the log
of A base B you're talking
about some exponent.
Right? So this is where you're
kind of multiplying exponents.
N is the exponent up
here, in the original.
Right? See it?
There's the original exponent
of N. And now you're multiplying
that by this other
exponent, log of A base B,
also represents an exponent.
Remember, logs are
really exponents.
So that is the power
property of logs.
So here we have it.
The log base B of A to the N
is equal to N times the log
of A base B. Now
just be careful.
Log base B of A to the nth.
You can't simplify any further
by pulling the N
out to the front.
It's only if of the A
is on this base here.
I'm sorry, if the N, the
exponent, is on the base
of A. Okay, in this cautionary
one, you would actually have
to compute the log and then
raise it to the nth power.
All right, so what we're going
to do is take this property
and rewrite some problems.
But first, let's just
look at an example
of why it makes sense
to be true.
Let's look at this example.
Of the log base 2 of 4 cubed.
All right, well, I could simply
compute for cubed, which is 64.
Because that's 4
times 4 times 4.
And then the question would
be the log of 64 base 2.
6 to the what power equals
64, and that's actually 6.
So I've got that the log
base 2 of 4 cubed is 6.
Now let's compute this
using the power rule.
So I have the log
base 2 of 4 cubed.
What I could do is take
the 3, this exponent.
Put it out in front,
times the log base 2 of 4.
Now it's actually
simpler for most people
to compute the log of 4 base 2.
So I get 3 times -- right,
what's the log of 4 base 2?
Well, 2 to the what power is 4?
It's 2. So note you get
the same answer of 6.
So this is just an
illustration of one example
where you could use the
power property of logarithms
and get exactly the same answer.
So here are three
problems for you to try.
Use the power property of logs
to rewrite each expression.
Go ahead and put the video on
pause and try these on your own.
All right, let's
do the first one.
All right, so we simply
can bring this exponent,
this 5, out to the front.
And multiply it.
So we have 5 times the
log base 3 of X. Next one.
Well, you don't see this as an
exponent, it says the cubed root
of M, but remember what
that means, the cubed root
of M. It means M to the 1/3.
So if you want, you could
rewrite that as M to the 1/3.
And then you can bring the
1/3 out to the front as 1/3.
Log base 4 of M. All
right, and this last one.
We move this down a little bit.
We've got a negative 1.
So that will be negative
1 or just a minus sign.
Log base of 12.
Right? You could also write that
as negative 1, that's up to you.
Now by the way, just
keep in mind what 12
to the negative 1 really
means, it means 1/12.
So when I write 12
to the negative 1
that's a different way
of writing the number 1/12.
Here are a couple
for you to try.
You want to write
these as single logs.
So you don't want to
have a number in front
of the log, in other words.
Okay, try those by
putting the video on pause.
So what do we have here.
I need to move the 3,
right, to the exponent on 2.
So there will be log
base 5 of 2 cubed.
And then we have to
simplify 2 cubed, which is 8.
So there's our answer.
Second one, you put the 2
on the exponent of the M,
solve the log base
4 of M squared.
All right, to review, basically
what we've done with logs
so far, we have the
basic definition
that if A is greater zero,
B is greater than zero,
and zero is not equal
to 1, then the log
of A base B equals N
means B to the N equals A.
And then we have these three
basic properties, the log of 1,
no matter what the
base is, unless it's 1,
of course, is equal to zero.
Because B to the
zero will equal 1.
The log of B base
B is equal to 1,
since B to the first
power equals 1.
That makes sense.
And the log of B to the N base
B equals N. And then we went
over three properties.
The product property, the
log of X Y base B is log
of X base B plus the log of Y
base B. The quotient property.
The log of X over Y
base B equals the log
of X base B minus the log of Y
base B. And the power property,
which is the log of A to the
nth base B, is N times the log
of A base B. All right, we're
going to be working with all
of these properties
together in the next video.
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