In this video, we are going to talk about
properties of logarithms.  So if M, N and
a are positive real numbers with a
not equal to one and x is any real
number, then so let's go through each one
of these on an individual basis.
So let's start with the work for
number one.  So we've got log base
a a is equal to well let's call this
question mark.  Now I can write this as an
exponential. So that's a to the question mark is equal to a.  Well the only thing that
makes sense in this particular case is
question mark is equal to one.  So log base a of
a is equal to one.
Okay now let's go for number two.  The work for number two, it's saying log base a
of one, and again will put equals
question mark.
Similarly to number one, I can write as an
exponential, a to the question mark is equal
to one. Now the only time that is a
true statement is when question mark is
equal to 0.  So log base a of one is equal to zero. Now let's go through the
work for number three then.  It's saying
log base a of (a^x).  Again we will put
question mark and again similarly to
the previous two, I am going write it as an
exponential.  So that's a to the question
mark is equal to a^x.  Because
this is a function, we know there's only
one answer and the question mark in this
particular case is x.
So I'm going to scroll back up and fill in the first 3 answers.
Log base a of a is one, log base a of 1 is zero
and log base a of (a^x) is equal
to x.  Ok now I'm going to go to the next
page then, again I'll come back and fill
in the rest of these, but let's go
to the next page and do work for number
four then.  Now for number four, remember we
know that f
composed with f inverse of x is equal
to x and f inverse of f(x) is equal to x
because they're inverses of each other.  So let's do this then.  Let's f(x) is
equal to a^x and g(x) is equal to log
base a of x. So now I am going to do the composition
f(g(x)).  Well according to what I've
written, g(x) is log base a of x.  Now
remember every time I see an x I will 
replace with log base a of x. So then
that is now this.  Again because a^x
 and log base a of x are
inverses of each other, I know this
particular answer is going to be x.  So now
know that. Now let's talk about an example of number five.  Lets do log base 3 of 81.  Now
we know log base 3 of 81 is the same thing as log base 3 of 27 times 3 because 27 times 3 equals 81.
Now what I would like to do, is to
separate those two logs.  So I'm going to put log base 3 of 27
and I'm going to put log base 3 of 3.  Ok now, log base 3 of 81, well that's in another words
I will write on the side here, three to what
power is equal to 81?  That's four, log base 3 of 27,
that's the same thing as saying three to
what power is equal to 27.  We know that
is three and we just finished talking about
log base three of three, we know that's one.  So
now, what basic operation can I put in
between those logs to make it a true statement?
Well that's addition here.  So when I have
the multiplication of logs such as this,
again I will go back to the original and
rewrite.  If its multiplication, I can
separate them with the addition.  Again
notice, the bases must be the same in
order for this to happen.  Just like when
you multiply exponents, you add the
powers.  Ok, so let's do an example of
where its division to start with. So I'm
going to do now,
log base 2 of 32.  Well that is the same thing as log base 2 of 64 over 2 because 64 divided by 2 is 32. Ok,  I want to
be able to separate this.  Ok,
so we know 2^5 = 32, 
so log base 2 of 32 is five.
2^6 is 64 so log base 2 of 64 is six and we
already know log base two of two is equal to one.  So now
similarly to what I just asked a minute ago, what basic operation can I put in
between six and 1 to equal 5?  In this particular case, it is subtraction. So if I have a log
with division and I want separate this, then I know it's going to be subtraction in
between and again keep in mind, the
bases must be the same.  Ok let's do
something similar for number seven.  I'm
going to write out literally the original
part.  Now an exponent is nothing more
then repeated multiplication.  So I can
write out this.  M times M times M... M
p times.  Now we just figured out from
two examples ago, when it's
multiplication, I can separate them
by addition.  So it is log
base a of M + log base a log  of M + ... + log base a of M
but that's p times.  So when you're
adding the same exact thing p times, it
ends up being p on the outside
log base a of m.  Now let's go back to
first page for this and fill these
in then as such.  So first of all, for number
four, a taken to the log base a of x is x,
for x greater than 0, domain restrictions.
Log base a of MN, is the same
thing as log base a of M + log base a of N.  For number six,
log base a of (M/N) is the same
thing as log base a of M - log base a
of N.  For number seven, log base a
of M to the P power is the same thing as P
times log base a of M.
Ok so all of these properties are nice to
know, but we will be using a lot of 5, 6, and 7 for
later videos and we'll take a look at
the first 4 more in this video.
Ok so I'm gonna skip to these third page
now, and now I want to be able to
simplify the following.  So log base 69 
of 1.  Well that is similar to property 2 where
log base a of 1 is equal to 0.  So log base 69 of one is zero.  The common law of 10, remember that's
the same thing as log base 10 of 10.
That's very similar to property one, where
log base a of a is equal to 1.  So the common log of 10 is one.  Log base 7 of (7^3), that's very very
similar to the third property, where
log base a of (a^x) = x.
so in this case, it's three.  Now this
last one's a little bit more involved
for this.  It's using property four, but it's not
quite there yet because of the 2 that is
on the outside.  But now, I can use
property 7 because I have a number
on the outside I could put it back
inside the log.  So I can rewrite it as this.
We know that's going to be 25, and now I
can use property four directly, 12 taken
to the log base 12 of 25,  that's equal to 25.
Now because
natural logarithms are special types of
logarithms, I know these particular
properties quickly also.  So for instance,
the natural log of one,
well again, that's similar to log base a of 1 that's equal to zero.  So that's zero.
The natural log of e, well again
rewriting it is the natural log base e of e
that's very very similar to property 1, that should equal one there.
Well, the natural log of (e^x), that's similar to property three, where its log base a of (a^x)
is equal to x, same thing here,
and because e and the natural log are inverses,
I know this fourth one is x also.
Ok, so now I want to be able to simplify
these two.  So the natural log of e^5,
well again, from property three,
we know that's five.  So e taken to the
natural log of three, because again e and the
natural log are inverses of each other, I
know this answer is going to be three.
