HELLO, Mr. Tarrou. We are getting our logarithms,
we graphed logarithms, we introduced the basic
structure of logarithms, how when you take
the log of a number you do get an exponent.
Such as the log of a hundred is two and the
reason is if we just write the word log with
no base indicated that is base ten. The log
of 100 is 2 because ten squared equals a hundred.
We also talked about how logarithms are the
inverse of the exponential function. You log
a hundred you get two and if you ten to the
second power you are back to a hundred. At
any rate, we want today to talk about expanding
and contracting. We are just going to manipulate
log expressions, getting us ready algebraically
to start solving equations with the variable
in the exponent. Which will be my next video
or the next couple of videos. So, we need
to learn how to take apart or sometimes put
back together these log functions to aid us
in next section that we are going to take
a look at. So no solving equations today,
just manipulating expressions and occasionally
simplifying some logs along the way that we
can do in our head. Log base b of m times
n. I want to rewrite this expression so that
the M and the N, each of those factors have
their own log function. Well inside of this
multiplication, inside of this log function
you have a multiplication that has a common
base of b. Remember, what do you get out of
a logarithm equation or a log function? You
get an exponent. So the properties of logarithms
are going to be very similar to when you worked
with like bases and you were dealing with
exponents. Such as, x to the fifth times x
squared. Well, when you multiply like bases
you add the exponents. Out of a log function
you get an exponent and the multiplication
within this log has the same common base of
b, so if I want to separate this multiplication
and write log base b of m and log base b of
n what I am going do with that? How am I going
to separate that? From a log function you
get an exponent. From a log function you get
an exponent. When you have multiplication
within an common base, very similar to your
rules with regular exponents, you add them.
When you multiply with like bases you add
the exponents, so there is the first property
of logarithms. Again, based off of what you
know already about multiplying like bases
when you are in exponential form. What did
you do when dividing like bases and you are
learning about exponents, you were taught
to subtract. So when you have division and
a like base you subtract the exponents. Logarithms
give you exponents, so the log base b of m
divided by n is equal to log base b of m minus
the log base b of n. Again very similar to
when you are dividing with like bases in exponential
form. Now again, this might go against our
idea of order of operations, but within a
logarithm when you do the log of a number
you get an exponent. So this is kid of like
an exponent raised to another exponent. Like
if I were to write x to the fifth raised to
another power of two, where these exponents
are stacked. We would drop that down and write
x to the two times five power or x to the
tenth power. Well that is the power of logarithms
we are looking at here. It is going to allow
us...Logarithms are going to tell us if you
take the log of a number that has its own
exponent, you can drop that exponent down
out front and write it as p times the log
base b of m. Again, think of is at, well it
is not as it is true, when you take the log
of a number you get an exponent so this is
the power to power property that you would
see when dealing with normal exponents in
exponential form. Those stacked exponents
will drop down and you multiply. So, you can
do the same thing with logarithms. So there
is our three properties of logarithms. I will
probably need to erase that at some point
to have room for a couple of my examples.
Let's start taking some log expressions and
breaking them apart, expanding them. So our
first one is going to be log base five of
5x. So it is the log of two items multiplied
together. Well, when you take the log of two
items being multiplied together with a common
base you can expand that by doing log base
five of five plus log base five of x. Multiplication
with a common base, you add those exponents...
you add those logarithms when you pull it
apart. Now occasionally we are going to have
some parts of these expressions that can be
simplified. Five to what power will equal
five. When you log five with a base of five,
again when you log a number you get an exponent,
so five to what power...what exponent...gives
you five? One plus log base five of x. So
we got done expanding that one. Next! We have
got log base three of twenty-seven x squared.
Same thing, very similar to the last question.
It is the log of two items being multiplied
together. So we are going to expand that multiplication
and we do have a power of two but we are not
going to be able to bring that down out front
until we have it exactly in this format of
log with a leading coefficient of one and
a single item with that exponent. Basically
when we are expanding our log expression,
you are going to deal with this power to power
rule last. So this is going to be log base
three of twenty-seven plus the log base three
of x squared. Alright...now again when you
log a number you get an exponent and it is
an exponent of this base. Three to what power
is twenty-seven? The log base three of twenty-seven
is three, nine, twenty-seven. Ok, now we have
three plus the log base three of x squared.
Now we have that leading coefficient of one,
we are taking the log of a single factor that
has an exponent...that power of two. That
power can now be dropped down out front and
we get three plus two times log base three
of x. That expression has now been completely
expanded and evaluated a little bit with the
log base three of twenty-seven. Don't worry
about working this does not come out to be
a whole number. At this point in your textbook
you should be given logarithms that can be
done in your head without the aid of a calculator.
If they do not come out evenly, then yes you
will have to use a calculator. Log base five
of the square root of x over 125. We are going
to deal with square root in a minute. You
should really not work with radicals in that
radical form. They are actually much more
complicated. You should rewrite them with
parenthesis and us a fractional exponent where
the top is the power and the bottom is the
root. I will go over that in just a second.
The first thing I want to do is split up that
division. Division signs do act as grouping
symbols not there is too much going on with
this fraction. We want to split that big fraction
bar first and then take care of the top and
the bottom separately. My students that do
tend to make a little bit of a mistake on
these problems are the ones try to do more
than one step at a time. You really want to
do these log function problems piece by piece
by piece. When you try to do too much in your
head, it will usually lead to some careless
mistakes. So it is the log of the numerator
minus the log of the denominator. Again there
is a power of five in 125. This will work
out evenly. Five to what power is going to
give you 125? What exponent will you get out
of the logarithm function? 5, 25, 125... so
that will be three. Now this looks like it
is done, it looks like it is expanded as much
as possible. But if I write this radical symbol
with its index of two and the x having a power
of one that is going to be log base five of
x raised to the one half power. Well now I
am taking the log of a single factor that
has its own exponent and that is again your
power to power rule when you are dealing with
exponential form. You are allowed to bring
that one half down out front. This is the
property that is so handy when we start to
solve equations with the variable in the exponent.
That is why logarithms are so handy. Now are
not doing any of that today. OK...BAM! Got
that done:) Fully expanded...moving on:) I
have not done any natural logs yet. So let's
take a look at that. We have got the natural
log of e to the third over seven. Don't be
distracted by this Ln, this is just another
notation or short hand for log base e. I will
expand that out as a teaching tool here in
a second. But again first I want to split
up that sign of division. So we have got the
natural log of e to the third minus the natural
log of seven. Now, now there is a couple of
things I can do right now. I can pull the
three out front with this power to power rule
thing but I want to show you something different
going on here. Natural log, this is standard
short hand and you should get used to it,
but natural log is log base e. Then it is
log base e of e to the third power minus the
log base e to the seventh. There is no reason
to really write it like this except I am trying
to highlight for you that ln is log base e.
E is approximately 2.718 and that number shows
up a lot in nature and engineering fields
and it is approximately 2.718 again. It is
kind of like pi. Ok, we have this log that
has a base and we are logging a number that
has the same base. I explained in an earlier
video, the introductory to logarithms, why
they do cancel out...and they do. So the log
base e and logging a number with it's own
base of e, those are going to cancel out and
the three is simply going to drop down. I
am going to back to the Ln because this is
not...it is a good teaching notation but no
one really uses this. The natural log of seven.
Alright...done! Let's do a couple more. I
think I am going to have to do a separate
video for condensing these expressions. Because
I am sure I am getting close to running out
of time. Log base b of x to the fourth times
y squared over z. Another question where there
is a lot of division and multiplication, etc
going on. I am going to split up this fraction
bar first. So we have log base b of x to the
fourth times y squared minus log base b of
z. Great... So I have not worried about splitting
this all up at once and sometimes you can
do that and it is correct and sometimes you
will do that and you will miss a critical
step and cause a minor error. I am just showing
the expansion of the division bar with all
of the numerator and all of the denominator.
I am not trying to do two steps at once. Now
we have the log of two factors being multiplied
together and the fact that there are two items
here with there own exponents which also why
I am not pulling the four and the two out
yet. So we get log base b of x to the fourth,
splitting this multiplication up and giving
each factor their own logarithm, will be plus
log base b of y squared minus log base b of
z. And now that I have these terms completely
isolated I can drop down the four and take
care of that power property last. And now
that expression is fully expanded and that
is pretty much the last example I wanted to
do. I will do one more expansion in the next
lesson because I want to show one where there
is a square root over all this. I am Mr. Tarrou.
I am out! Thank you for watching:) BAM!
