- [Voiceover] Today, we
are going to take a look at
Properties of Logarithm.
First property, log x-y with the base a
equals log x with the base a
plus log y with the base a.
Let's prove the log property #1.
Let log x with the base a equals m
and let log y with the base a equals n.
Let's convert both of them
back into the expotential form.
So we have a to the m equals x,
we have a to the n equals y.
Now we multiply both equations together
so we have a to the m
times a to the n
equals x times y.
On the left-hand side,
we have two expotential
form times together
share the same base.
So we can simplify that,
we keep the base and we add the exponent.
Now we have a to the m plus n
equals x times y.
Finally, we want to convert
this back into the log form.
So we got log x-y with the base a
equals to the exponent m plus n.
But what is m?
M, indeed, is log x with the base a.
And what is n?
N is log y with the base a.
So now we have the property #1.
We have the property #1.
Let's take a look at
property #1 in detail.
Be sure we remember properly.
We have the log of a product
equals sum of the log of each factor.
So when we have log a-x with the base 3,
we can expand this
make it becomes log 8 with the base 3
plus log x with the base 3.
When we have log 7
times 3 with the base a,
we can expand it,
make it becomes log 7 with the base a
plus log 3 with the base a.
Remember the property in a proper way.
Do not try to remember
half of the sentence.
Sometimes we would tends to think
whenever we have a product, we add.
That would becomes very scary.
We have careless mistake
like log 7 times 3 equals to log 10.
We have 7 times 3 equals 10.
Those are very scary mistake.
Let's see property #s 2.
Log x over y with the base a
equals log x with the base a
minus log y with the base a.
So now we have a log of a quotient
which is equals to log of numerator
minus log of denominator.
So when we have log 8
over x with the base 4,
that would be log 8 with the base 4
minus log x with the base 4.
When we have log 7 over 2 with the base a,
we can expand that,
make it becomes log 7 with the base a
minus log 2 with the base a.
Once again, remember the
whole properties in detail.
Do not think of whenever
we divide, we minus.
That would becomes a serious mistake,
would becomes a serious mistake.
Property #3.
Log x, raise it to the
y power with the base a
equals y times log x with the base a.
So we have log of x raised to an exponent.
That would becomes that particular
exponent times log of x.
So when we have log x cube with the base 5
we can bring the exponent
all the way to the front of that log,
making it 3 times log x with the base 5.
When we have log 8 raised to
the 12th power with the base 7,
we can bring the 12 all the
way to the front of that log
making it 12 times log 8 with the base 7,
8 with the base 7.
One more property, property #4.
Log x with the base a
equals log x with the base b
divided by log a with the base b.
Sometimes we refer this to
Change of Base Property.
That would be very helpful
whenever we need to change base.
So we have logs 12 with the base 4,
we can turn it into
log 12 over log 4.
Notice that on the right-hand side,
we are using common log.
So we change the base to 10
or we can have log 25 with the base 3,
which is equals to the natural log of 25
over the natural log of 3.
Notice that the base this
time would be e, would be e.
In summary, we have four properties.
Notice that the first three properties
always bear the same base.
So sometimes, we will simply say
log x-y equals log x plus log y.
Log x over y equals log x minus log y
and log x to the y power equals
y times log x.
And the last one is
Change of Base Property.
We can change the base
to whatever we like.
Most of the time, we select the base 10
so we have a common log
or we select the base as e
so we have a natural log.
E so we have a natural log.
