Assalamualaikum warahmatullahi wabaraktuh
In this video, we will discuss the laws of
logarithm and how to change its base.
Just like in Indices to apply
the rule of log you need to make sure
that they all have the same base.
Let's see the first law here if I have
log + log with the same base
it will become multiplication.
For example
log(5)_a + log(6)_a
First thing we need to do is to make sure they have
the same base and since both of these
terms have base a
this one can be log(5x6)_a
so this will become log(30)_a and as
usual I will leave this second example
here for you to do. I'll I'll tell you
when to pause the video.
The second rule is just like the first
rule.
If addition gives you multiplication, subtraction gives you division.
just like this one since they
both have the same base here.
This will become log(10/2)_a which can be simplified to be log(5)_a.
Moving on to the third rule over here.
If you have power for the number here we
can bring forward the power as its constant.
Always be aware that this is not
equal to this.
For this one the power belongs to
its number
for this one this power belong to whole log term
and you cannot bring forward this power for this one.
Do not be confused by that.
Let's look at the example over here.
We have log(1000)_a we know that a
thousand
can be written as 10^3 base a
and since this power belongs to
the number and not the whole log term
we can bring forward 3 as its constant.
So this can be written as
3 log(10)_a
Take time to complete the second example
over here.
Finished.
Let's check your answers.
The first one is ln(3xy)
The second one is
ln(2x/y)
and lastly
this one is equal to
5 ln(y). Easy, right?
Let's proceed. Rule number four is for
when we are changing base
we can change base a here as any base.
Say b.
It will become log(M)_b / log(a)_b
The number will be in its numerator
and the base will be in its denominator
For example here.
We want to change log 7 base a into base
5.
So this one will become log(7)_5 / log(a)_5
Let's look at rule number five.
We can also swap the number
with its base by making this to be its
reciprocal. See log(M)_a becomes 1/log(a)_M
For example, log(11)_a
we can change this into base 11 by
making it its reciprocal.
So 1/log(a)_11
Rule number six if base of index is equal
to base of the log like this.
We can cancel this out and left with just x,
for example, 10^log(x). As I said earlier
when the base for this one is not written.
We know that that this belongs to base 10. Since base of the index is equal to
the base of the log here both of them
cancel out and left us with just x and
lastly rule number seven. If the base and
the number are equal then this is equal to 1.
For example log(4)_2. This 2 is
not equal but we know that 4 is equal to
2^2 and we can rewrite this one as
log(2^2)_2
and by rule number three
we can bring forward this power over here.
So this can become 2 log(2)_2
now that these two are equal.
This will become 1
so 2 x 1 this is just 2.
Now it's your turn to do this example on
your own. Pause this video now and then
we'll check your answers. All done
for row number four, the answer is log(x)_2 / log(3)_2
Next your answer should be
1/log(x)_y
Rule number six.
This one is equal to 2x and lastly for
row number seven
This is equal to log((4)^-1)_4 and
this is equal to -1.
Now we know this laws.
Let's try to do some questions.
As I said before similar in index.
We need to make sure
that they all have the same base.
This one has base 8 and this one has base 2.
You can either change this one into
base 2 or this one into base 8.
In this solution.
I'll choose base 2 that means I need to
change this into base 2. So changing
this into base 2 becomes log(32)_2 / log(8)_2  - log(8)_2.
Now that all in the same base.
We can simplify.
We know that 32=2^5 and we know that 8=2^3
So this one will become
log(2^5)_2 / log(2^3)_2 - log(2^3)_2
and by rule number 3 we can bring forward its
power over here.
So this can be
5 log(2)_2 / 3 log(2)_2 - 3log(2)_2
and we know that by rule number seven this
this and this are equal to 1. So this will
become
5/3 - 3.
So this is equal to -4/3.
Now.
You can try this question on your own try
to simplify this as one log term.
Pause this video now and we'll discuss
the answer after this. Finished.
Let's check.
We know that all this already in the same
base.
These are all in ln
that means they all have base e.
Next we are going to combine them as only one log term.
To do that.
We need to bring all this constant
back up as its power.
So this will become ln(7^2)+ln(8a)-ln(a^2)^(1/2)
by rule number one and two
addition will become multiplication and
subtraction will become division.
So this will become ln((7^2)x8a)/(a^2)^(1/2)
Simplify more you're going to get ln((49x8a)/a)
we can cancel this a and 49 x 8 will be
ln(392)
and that will be your final answer.
So that's it for now. In the next video.
We will discuss how to solve the equation
of log and equation of indices with
different bases.
Thank you for your attention.
Have a great day.
And take care.
