Hello, welcome!
My name is Meghan, I'm one of the
instructional assistants that works in
the Academic Support math center on the
Lee campus.
Today I want to talk about some rules
about logarithms.
In particular, after I go over these
general rules, I like to do some examples
of
condensing and expanding them. Now logs
are going to be particularly helpful
once you get to certain courses.
I find that mastering these rules now
will really help you in some of your
future work.
So first things first; let's actually go
over some rules when it comes to the
exponentials.
So when it comes to your exponents the
following rules apply:
whenever you see
something like
x to the second times x to the fifth
that shows that really you add the
exponents it's really like saying x to
the seven.
Your next problem if you had something
like x to the fifth
and you were dividing it by x to the
second
you would subtract the exponents so in
this case x to the third.
Whenever your variable's power
is external, like this you're multiplying
them. So 2 times 5
it'd be x to the 10. Alternatively,
if you have a fraction exponent, like
perhaps you have
x to a three halves.
Well, we're actually going to set up both
a radical and a power.
The denominator goes here and the
numerator goes here.
But it should be noted whenever that
radical has a 2
that's your standard square root. So you
can write it like so.
And then finally whenever your exponent
is negative,
it become positive by changing its
position on a fraction.
So if it's a normal value like that put
it in the denominator like so and you
now have a positive exponent.
I just wanted to bring up these rules
because they're going to help us with
some of the stuff we'll be doing later.
There we go. So I first also want to go
over
the logarithmic form versus the
exponential form.
Now you might sometimes have something
in one form that would be better written
in another form.
Converting between the two is not too
bad. You just need to identify
which parts go where, and then those same
things go on the other side.
So like as you see here log base b
m equals y. That same b
is used in both places here. that y
goes here and that m goes here.
So let's say you have something like log
base 8 x equals 64.
So that would be like if we wanted to
convert it.
8 to the 64
equals x. That would be a very big number.
But the point is we can convert between the
two.
Now let's go over some properties of
logarithms.
In many ways you can say this relates
pretty well to those exponentials that
we were just looking at.
But often you'll be asked to work with
logarithms, so it's really important to
know these rules.
So keep all these in mind and feel free
to pause at this point if you need to
take notes on these rules because we'll
be using them to help us with the next
two problems.
Here we go. So for our first example we
want to expand this log as far as we can.
And there's going to be quite a few
steps to what we're doing here.
So the first thing i personally like to
do
is check if there's any fraction
occurring. And it does seem to be
happening.
There does happen to be one right here.
So when I write this
I'm going to employ
the rule that whenever log
base b has a fraction,
this is the same as your log
being subtracted.
So we'll be doing that part first so we
have log base 3
of your first term which would be
everything in the numerator.
Minus the log base b of everything in
your denominator,
like so. So that's our step one.
The next part I like to do is look to
see if there's any products being
multiplied together,
because remember whenever they're being
multiplied,
we can add the two logs when they are  separate.
So I'll be doing that here.
We have log base 3
of 81 plus
log base b of x squared
minus now, this minus applies to
everything in the denominator so I
recommend having a parenthesis out here,
to show the minus still applies to it.
And we have our log base b of y to the
fourth.
And our log base b that's going to be
base three.
That's a silly mistake let me just see
where I said
Sorry about that it's easy to lose track
your terms.
There we go, our next step is to
check the powers of what we have. So I
want to put everything
in a most simplified exponential form.
When I look at my 81 I can actually
rewrite that
as 3 to some power. In this case
81 is the same
as 3 to the fourth. And I want to see
powers of three,
due to the fact that I have a base of
three. Because I will be using
that log base b
whenever this matches the same as the
base.
It simplifies whatever your exponent is.
So if 81 is the same as 3 to the fourth,
that means this entire first part is now
just a 4.
Additionally, I'm also going to use the
rule
that whenever you have an exponent
you can move that exponent in front.
So I'll be doing that as well since x
squared,
that means an exponent of 2, I'm going to
place that 2 in front.
This log has an exponent of 4. And this
z,
recall that whenever you have a radical
you can write it as an exponential fraction.
So that means I'll put a one fifth in
front of that term,
giving us a 2 log
base 3 of x. And we also have this minus
sign in front,
so we might as well distribute that. So
we have a minus
4 log base 3 of y
minus 1/5 log
base 3 of z. So
there are a lot of different logarithm
rules which can all be used even within
the same problem.
So it's definitely good to practice
those rules and have them known for
whenever you're dealing with logs in
your classes.
Now let's look at a problem that's
actually the
exact opposite of what we just did.
Instead of expanding our log,
we want to see if we can condense a log.
So we're going to try to put this all
into one term.
When I work in this order last time, the
final step I did
was putting all those exponents in front.
And I'm going to work in a reverse order
here. So I'm going to put all those
exponents up.
So I begin by taking each coefficient in
the front,
and putting it as an exponent. So this
one,
since the coefficient front is 2 it's
going to be log
base 2 of three squared.
Our next term we have log base two.
And instead of saying x to the one for
power, remember that a fraction exponent
is really just a radical.
So I can write it like so.
Then our next one is log base 2 of y to
the fourth.
There's no coefficient in front of our
last one, so we can keep that one as is.
From here I noticed that our two
end terms here have a negative in
common.
So I'm going to factor that negative out.
Rewriting, we have log
base 2 and 3 squared. It's fine for me to
just write it as a 9.
And now I'm going to take that common
negative out, leaving log
base 2 y to the fourth plus
log base 2 z.
Now at this point I notice that we have
a couple terms being added together.
So this is where we can use that the log
of M plus the log of N,
is the same as one log with
M and N multiplied together. So I'll be
doing that for both groups.
We have log base 2 and then we have 9
times cube root x
minus log base 2
of y to the fourth times z
So now my final step is to look at
whenever you're subtracting logs.
This is the same as dividing them as one
term.
So I'll be doing that here. We'll have my
first term as the numerator.
And since we're subtracting it by the
second term, that's your denominator.
So our final solution is log base two,
and then your numerator is nine times
the cube root of x,
all divided by y to the fourth times z.
So that is a fully condensed logarithm.
So thank you for sticking around for
this video. If you have any
questions, whether it is on logs or
anything else in your class,
feel free to schedule a tutoring
appointment with the Academic Support's
math center.
We offer individual tutoring but we also
have a walk-in basis.
Just feel free to look for more details
on our site,
and thank you again. Have a good one!
