Hi everybody! So we're gonna go over the
proofs of the properties of logarithms
and a lot of these proofs are gonna
depend upon us being able to move back
and forth between exponential form and
logarithmic form and so just a quick
reminder. The exponential form of an
equation would be B raised to the M is
equal to X and then that exact same
expression written in logarithmic form
is the log base B of X is equal to M so
the log base B of X is equal to the
exponent that you would get in
exponential form.
Just keep in mind these two equations
are the exact same equation they are
just written in different forms. So now
we're going to go through the proofs of
these of these properties. The first
proof that we're going to look at or the
first property that we're going to look
at is the one about a product. So the log
of a product is the sum of the logs so
log base B of X times y is equal to the
log base B of X plus the log base B of Y
and the way that we go about the proof
is that we're going to let the log base
B of X be equal to M and the log base B
of Y be equal to n and then I that's me
writing it in logarithmic form and then
I'll rewrite both of those things in
exponential form so again log base B of
X is equal to M means that B to the M
power is equal to X and that's what you
see right here and then similarly B to
the N power is equal to Y and that's
what we see right here. So we're gonna
start by looking at the product x times
y and since x and y are both equal to B
to the M and B to the M respectively, I
can write it as B to the M times B to
the N and then this is a product and I
learned in my algebra class that I can
add those two exponents together so I
can write it as B to the M plus n so I
have x times y is equal to B raised to
the M plus N and I can think about that
as being an exponential form and so I'm
going to write it in logarithmic form
which would be the log base B of
times y is equal to M plus n so all I
did was rewrite this as in a in
logarithmic form and then if I
researched the toot for the values of M
and n M is equal to the log base B of X
and is equal to the log base B of Y I
see that I get that property so now the
log base B of x times y is equal to the
log base B of X plus the log base B of Y
that's supposed to be a Y right there
sorry
so therefore log base B of x times y is
equal to the log base B of X plus the
log base B of Y so that's the proof of
the first property and that may seem a
little rough the the first time that we
go through it but what you're gonna see
is that that this idea is going to be
what gets us the rest of these the rest
of these properties so the second
property of logarithm says that the log
base B of X divided by Y is equal to the
log base B of X minus the log base B of
Y so that the log of a quotient is the
difference of the logarithms and I'm
gonna start this proof the exact same
way I'm gonna let M be the log base B of
X and and be the log base B of Y which
implies that B to the M is equal to X
and B to the N is equal to Y that is the
exact same thing that I wrote at the
beginning of the previous proof and and
the next step in the previous proof was
that I let I started with x times y now
I'm gonna start with X divided by Y and
so I can substitute in those values of B
so X divided by Y is gonna be B to the M
divided by B to the N my algebra teacher
taught me that if I am dividing two
things that have the same base then I
subtract the exponents so that's B to
the M minus n which I can rewrite as x
times y is equal to B to the M minus N
and now this is in exponential form I
can write it in logarithmic form as the
log base B of X divided by Y is equal to
M minus N and then I will substitute in
the two things that I started with M and
so I get that property there's that I
must have copied and pasted that
so we're probably gonna see that typo
the rest of this video so log base B of
X divided by Y is equal to the log base
B of X minus the log base B of Y so
there's the first the proofs of the
first two properties next we look at the
proof of the third property the log base
B of X to the R is equal to R times the
log base B of X this is probably the
most important property the one that
gets used the most that says that if
you've got a log of something raised to
a power you can take that power and move
it in front of the logarithm we're gonna
see a very similar proof to the previous
two slightly different this time I only
need one logarithm so I'm gonna let M be
the log base B of X rewrite that in
exponential form B to the M is equal to
X and then I will start with that of
that equation in exponential form and
I'm gonna raise both sides of that
equation to the R power which is
something that I'm totally allowed to do
on the left side here I have an exponent
I have a base raised to a power raised
to a power my algebra teacher taught me
that I would multiply those two powers
together so I could multiply them as to
get B to the R M is equal to X to the R
and then I'm going to rewrite that in in
logarithmic form
so the log base B of X to the R is equal
to R times M then if I reefs of stitute
in the value that I had for M over here
I would get that the log base B of X
city R is equal to R times the log base
B of X so that's the proof of the third
property therefore the log base B of X
to the R is equal to R times the log
base B of X the fourth property says
that if you've got a base and you have
an exponent whose and that is a
logarithm of the same base then it's
almost like all of this just cancels out
and you end up
with that that argument of the logarithm
the way to prove this it's this one is
actually the easiest of them but it's so
easy that it can be a little bit
confusing I'm gonna consider log base B
is equal to the log base B of X is equal
to the log base B of X I just wrote the
exact same thing on both sides of the
equation and if I think about this as
being an equation in logarithmic form
that is that a log base B of X is equal
to something and it turns out that
something is a logarithm but if I just
ignore that that it's equal to something
and I rewrite that in exponential form
it will be B raised to that log base B
of X power is going to be equal to X and
so rewriting that in exponential form
makes it be the thing that I ended up
having to prove right so that went like
I said that one's pretty straightforward
and that makes it a little bit confusing
all I'm saying is a log base B of X is
equal to the log base B of X and then
rewriting it in exponential form last
thing that we're gonna look at is the
change of base formula also a really
important one this is the thing that you
know most calculators just have a log
base 10 and a natural log of log base B
and this is what allows us to get any
type of a logarithm into one of those
two log bases so that our calculators
will work says the log base B of X is
equal to the log base C of X divided by
the log base C that should be a B wow I
got lots of typos here so that should be
a letter B there so I'm gonna start by
letting M equal the log base B of X and
then similar to what we've been doing
before that says B to the M is equal to
X and then I'm going to take the log
base C of both sides of this equation so
I get the log base C of B to the M is
equal to the log base C of X and one of
the previous proofs we proved that if
you've got a log of something raised to
a power we can move the power in front
the logarithm and so if I do that I get
M times the log base B a log base C of B
is equal to the log base C of X I'm
gonna divide both sides by this
logarithm to get M is equal to the log
base C of X divided by the log base C of
B and then what was M it was the log
base B of X so I get the log base B of X
is equal to the log base C of X divided
by the log base C of B again that should
have been a B right there so those are
the proofs of these very important
properties of logarithms that we're
going to use a whole lot and and what
these proofs do is justify our ability
to be able to use them whenever we need
them
