Now that we've seen what logarithms are,
let's look at how they behave and look
at the properties of logarithms.
The first property I'll go over,
I think you already know , is that for any
base b, the log base b of one is going
to be zero. Recall that b to the 0 power
is always one. So of course that follows.
Going backwards, the log using base b to
get one the exponent the log is 0. The
second property is that for any b, any
base b, the log using base b of b to the
X is X. Now, sometimes that's so easy it's
hard.
That's like saying who's buried in
Grant's tomb or what was Elvis's first name
or when was the War of 1812? You see how this is so easy it's almost hard.
It's like saying what is the exponent of
B to the X using a base of b. Well, of
course it's x. Now why is that
important, this second property here?
Well, this is going to allow us to solve
an equation like this one. Looks
difficult but in the future what we're
going to do...
we're going to introduce a new tool
taking the log of both sides and if we
take the log of this (think about it) on
the left side, the log base 2 of 2 to the
X is X. It's going to allow us to solve
some very difficult equations. Now that X
will equal the log base 2 of 7 whatever
that is but, and, and we'll figure that
out; we'll work with that later. So I hope
you see that second property is going to
be quite useful, that  the log base B of B
to the X simplifies to X. Look familiar?
Okay, let's move on to the other properties.
Basically, this is going to be divided
into three properties. This is the one you're
going to use way more often since
logarithms are exponents.
Remember the rule of exponents. When you
multiply expressions that contain
exponents you add the exponents. Well,
guess what? Logarithms are exponents so
when you multiply an expression that
contains a log you add the logs.
Another property of logarithms since
logarithms are exponents is that when
you divide expressions with exponents
you subtract the exponents, don't you?
Guess what?
When you divide expressions with
logarithms, you subtract the logarithms.
And finally when, we have one more
property of logarithms since logarithms
are exponents, when you exponent an
exponent, we end up multiplying them,
don't we?
Well, when you exponent a logarithm you
end up multiplying them. Look at the
other place where you can put the b and
this is going to be the most useful of
all the properties because that b is
hard to get when it's up in the exponent.
Another place to put it is down in front
and we can solve an equation much easier.
Get rid of the b, if you would, when it's
down on the ground.
Ok, so that's another location we can put
it.
Let's use the properties of logarithms
to expand this problem. We'll start with,
we have an expression here with division.
Logarithm with division becomes a
subtraction of the logs; division
becomes a...notice that it becomes an
easier operation. This is good; this is a
good trend. Now, what else do we have in
here?
We have multiplication of X. Ooops, not
exponents multiplication of logarithms
and of course it's the same rule to, to take
the log of two numbers multiplied we'll
add the numbers.
Now, what about the third rule? Where else can we put these exponents?
The exponent rule tells us that we can
multiply. Once again, the next easier
operation and actually this isn't a rule
of logs. If you think about it,
three log of a's is log of a plus log of a plus
log of a, isn't it?
We can bring that one down.
and minus 2 log of C's is ...so actually we
can take a very complicated calculation
and turn it into a bunch of pluses and
minuses and this is going to be very
important eventually because, because
that is the way your computer and your
calculator work. It uses logarithms to
change these very difficult problems
with exponents and division and
multiplication into a bunch of pluses
and minuses or ons and offs and that is
a fact. An expansion like this may be
interesting but for the most part we're
going to be using logs, or the properties
of logarithms to condense a problem.
We're going to work backwards and
contract these into one giant logarithm.
Where can I put this two? Remember where else I'm allowed to put it. And what
about the three?
He can become an exponent and what does
addition become if we work backwards?
Something the logs means multiply
and basically if you think about it 3
squared x to the third is nine times
eight. We've made this into one big fat
logarithm. Well, you're going to want to
do it in the future to solve equations,
believe me. This is called a contraction.
Okay, let's do another one,
I work with the exponent first
So a number out front can move to that
exponent spot and back and forth.
Now, remember what we do with subtraction.
Subtraction becomes division
and we've made this into one big fat
logarithm. It's important for you to know
that one log is going to be our goal.
What does this contract to? Don't let it
scare you. The log of c plus the log of a
plus the log of b plus the log of i plus
the log of n. This is a classic
mathematical problem. Remember what
addition becomes multiplication so this
long expression actually becomes, well,
log cabin!
Sorry about that.
Okay, let's do another.
I want you to get good at contracting.
Make them into one big fat logarithm.
Addition is going to become
multiplication 25 x 3 and subtraction is
going to become division. We could write
it as a fraction and 75 divided by 5 is 15 so the
answer here is the log of 15. Remember, we want to make it into one log eventually.
You'll see why when we go to do
equations.
How about this one? Let's see. Let's work
on our exponents first. I say, you don't
see any exponents but there will be.
Where can I put that two?
Yeah.
Now I have another number out front.
Don't let him scare ya.
He goes up in the exponent position and
we know we can have half as an exponent.
Now this one's going to go up in the
exponent position for the whole
expression inside parentheses. Now all
we're going to have to do,
oh, what, what is X to the one-half? Let's
write that in simpler terms. X to the
one-half is the square root of x, isn't
it? So you could write it that way as
well.
Remember that's a rule of exponents. Now
addition, now, (Quit crying!) addition becomes
multiplication and subtraction becomes
division
and you have changed this into one big
logarithm. Won't get any harder than that!
Okay, well I want you to go practice
these properties, especially the
contractions into one logarithm because
we'll be using that in the next show.
