Now we're going to look at properties for logarithms. You've had the conversion rule:
you need to be able to go from logarithmic form to exponential and back.
Common log if all you see is log x:
log x and you don't see anything here. It's understood to be 10, or if you have ln x
it's understood to be base e.
Recall these rules. Theses are your exponential rules.
Have a base raised to power times same base raised to a power:
remember I can simplify by keeping the base and adding the exponents.
Base raised to power being divided by same base raised to power. I simplify by keeping my base and
top exponent minus bottom exponent.
If I have a base raised to a power and that in turn is raised to a power:  I can simplify
by keeping my base and multiplying my exponents.
Now properties of logarithms some of these you've seen.
Others you haven't so we'll be looking at those in just a second.
If I take log of 1 to any base remember my base has got to be greater than 0.
It's going to equal zero,
because if I rewrote this in exponential form I'd have a to the 0 and that would equal 1.
So if you press your log key on your calculator
and then press 1 it should give you 0; same thing if you press your ln key and
1 it will give you 0.
So if I have a log of a
number and the base is the exact same thing
remember this right here is an exponent of 1; it equals 1.
Now if I rewrote this this would be a to the first would equal a and
anytime you have a base raised to the first power you get that base right back.
So press log, l o g, on your calculator and
type in 10,
because remember
log is base 10, and then if you hit enter it should equal one.
You've seen this one. We've had this one.
Your base here is the same as the base here whatever your exponent is that your value here.
Product rule:  the log of a product
to a base is equal to the sum of those factors to that same base.
Think of your old
multiplication symbol, remember it was an x if you rotate it a little bit
you'll have a plus sign.  It's a way that you can remember a log of a product
goes to a sum.
Quotient rule:  when I have the log of a value being divided by another value I
can subtract those two.
See the sign right here separating?  Now remember I've got a little
subtraction right here.  Power rule if I have a log of a number raised to a power to whatever base,
that exponent I can move it out front.
So this is my power rule.  So I've got product rule, quotient rule, and power rule.
Now I've got one proof here for number 4; my product rule and
you know you haven't seen any proofs in here, so let's look at this proof and
this is what math people always do with proofs. They always start with a let and that means they're about to set you up.
So if I let log u base b equal x and log v base b equal y.
Well,
if I rewrote this
using, you know, I can rewrite these as exponentials.  I can have u equasl b to the x and
v equals b to the y.  Right?   Going from logarithmic form to exponential form.
Now
next part well, what if I decide to do u times v?
There's my u times v.  Well if we do u times v - well
that's the same thing as b to the x times b to the y.
Remember I can simplify this since my bases are the same that would be b to the x plus y.
Okay, next part.
Well, what if we wrote this last one into logarithmic form I would have
my x + y , okay,
equals log (uv) base b.
Let me move this down just a little bit.
I'll just delete it there.
Okay, here's my x + y = log (uv) base b,
but remember - remember at the very get go what x was and what y was.
So with substitution, I'm going to put log u base b here and here I'm going to put log v base b, and
that's a proof.  Now the other ones can be proved the same way,
but if I have the log of a product I can separate it up as a sum of those factors of those logs.
BIG NOTE:
Notice big red note here. There is no
distributive property and logarithms this is not equivalent,
because this right here when I have addition is when these two are multiplied.
So if I have a log of a sum or a difference; it has to stay the log of that sum or difference.
Now - Q-ProP:  what we're going to be doing with these properties is we're going to be
expanding and condensing and a lot of students have
some difficulties, because they don't know where to start.
So one year some years ago
I was reading in the newspaper and Sea Ray had this brand new propeller
and it was called Q prop and I thought that was cool, and I kept reading and then I went wait a minute
I could use this
for expansion of logarithms first thing we look for is quotients,
then we look for products, and the last thing we do is powers.  Okay?
So it says the quotient, product, and power rules of logarithms were used to rewrite if possible each of the following
logarithms and I'm going to look at these.  Now these are the examples of expanding and when I expand I go this way
I look for quotients. Then I look for products, and then I look for powers. Well the first problem right here.
I see a quotient.  Remember we separate with subtraction.
So I'm going to have log 6 base 4 minus log 7 base 4.
Now here. I've got a product. I've got a quotient, but I start with quotients first.  So the first thing
I'm going to do is separate numerator from denominator, so I'm going to have log (4p) base 3 minus
log q base 3.
Now I have to go back make sure I've got product.  I've got a product right here.
Products are separated by addition.
So I'm going to have log 4 base 3
plus log p base 3
minus the log q base three.  Now guys that's expansion.
Started with one single log, and we've expanded that to, you know, single values after each log.
Now here's one very similar to this one
and I usually tell my students don't trust a log that has a numerical base, and I'll show you why in just a second.
So first thing I see quotient; see quotient; so separate top from the bottom by subtraction.
Move this little fellow over.
Okay,
next thing I've got myself a
product right here. I separate by addition. Now remember this radical means a one-half exponent.
So let's see what I did right here.
Okay, see my one-half exponent. So if you see radicals, you're going to have to change them to
fractional exponents.  Now, this is why I tell my students don't trust a
numerical base.
Remember log 2 base 2
that equals 1. What if this was log 4?
Guys if you can rewrite that as 2 squared then this would be 2.  So I'm going to have a 1 here.
Exponents come out front. That's the last thing the power rule so that comes out front; so let's see how this thing simplifies.
So this is one; the one-half comes out front as a product; and everything else is written as is.
This one we don't have a rule that deals with addition/subtraction.
So this has to equal itself.  Remember we cannot distribute this through.  No! No!
This one, well I see a quotient. I see product.  I see powers.  So first thing:
quotient - let's go separate by subtraction and
notice I've got a z base, and I see no z's so that makes me happy.  Got myself, right here, a product
separate by
addition.
Powers - bring powers up front move them as
factors, as products.
So there it is and you can't tell me that that's not expanded from the original.  So if they ask you to it,
you know, expand the logarithm. You'll have a single one and boy you're going to have a real good time.
This must be fairly big and complicated first thing. I see a cube root. Well guys the first thing. I'm going to do here
that's one-third.
Remember, that's written as one-third
and then
with my rules I've got all of these taken to the one-third power, so distribute.
Now we're free to go here. I've got a quotient and
I've had products and powers before.
So first thing let's make this
turn this to a subtraction, separate numerator from denominator.
Next thing I have to worry about, remembering Q-ProP is products; so I've got a product right here, and they're separated by addition
and
now I've got little tiny baby powers that I've got to bring out front as
products, so there it is.
And guys if you're asked to expand you're going to also be asked to condense.
Now think about Q-ProP, if I go this way to expand does it make sense I have to go backwards
to condense.  So I'll be looking for powers remember where they're located - in front of the log.
Products - well if I see a product that means there's an addition, and then my quotients are separated when I've got subtraction.
So let's see what we've got here.
Got parentheses, so they want me to do this first.  Since I see subtraction this can be
singled down to one logarithm as log k over m base b. I still have that other one.
The other subtraction, so that will be this
k over m
divided by a
after my log base b,
but they're not going to leave this remember this is k divided by m
divided by a.
When you divide fractions remember you get the reciprocal and multiply so this would be k over m times 1 over a.
So that would be it.
Notice from more than one log to a single - single log.  So now these I see numbers out front.
So I've got to take those back in as
exponents.
Next part, let's see I've got
we can go ahead with
addition we can multiply.  Let's see if that's what happened next - nope.
First thing if you have a numerical base a lot of times
they'll go ahead and find out what that is.  So five to the third is 125; nine to the one-half
it's the same thing as square root of nine; so they went ahead and did that.
So now I
get to multiply these, because that's addition.
It's supposed to be a little time sign there.
If you multiply that will give you 375.
There you are.
Why don't you try this one?
What do you do first?
Hopefully, you said you take the exponent and remember I got to do Q-ProP in reverse.
Okay.
Now you might leave this as a one-half power, or you can make this as square root;
either one would be acceptable.  The next thing I've got subtraction, so it means a quotient and
that's it.  So from more than one logarithm; it's condensed down to one using the correct properties.
And that's all for this one. Thank you!
