Hello everyone, welcome back to a new video.
In today's video, we are going to do a topic which is
written on the board, that is logarithm.
We will do logarithm. In the last videos, we have done
indices and sets.
This topic is related to that. It is a little ahead of
that, logarithm.
If we see what we had read in indices, what did we read?
That there was a number and you had written some power of
that number and you had solved that power.
If it is a square power, then how many times we will
multiply the number? 2 times.
Similarly, if the power is 10, then the number will be
multiplied by 10 times and the answer will be in indices.
So, the number which you wrote below, which you used to
power, the number below was called your base.
And the number above, which used to be power, you can call
it power or you can call it index or you can call it
exponent.
All these will be the names of our power.
And a to the power x will have a particular answer, that
would be b.
Now, what is logarithm?
Logarithm is also a kind of operator or function.
So, how will we denote logarithm?
We will denote it with log.
Starting three letters, log and whatever we write in log,
that would be our argument.
What will be called as log argument? So, what is the use of
taking log? Let us see that.
So, the most basic example, I will explain you with an
example. You will understand it well.
For example, suppose we take 5 raised to the power 3 is
equal to 125.
5 into 5 into 5, that is 25 into 5, that would be 125.
Now, whose case is this? Your case of power and exponents,
indices, that 5 raised to the power 3 is your answer, 125.
Now, how will you do this in log? Suppose, what happens in
log? You have to find out this thing.
How? You will find out log of answer, which was your b.
You are finding out log of that.
So, you will find out log of 125.
So, whatever comes in the bracket of log, what did I tell
you?
We will call that as argument.
So, we will find log of 125.
Now, we will write base 5 in the bracket of log.
We will write base a in the bracket of log of that number.
Now, this value will come, which is your power, that is x.
And here is 3.
So, what does logarithm mean?
What do we have to find out? What is the question?
Log of 125 to the base.
How will we read this? To the base 5.
What will this be?
Indirectly, it means, what power of 5 should we raise?
That is, what power of base should we raise?
So that we get the answer, 125.
This is called our log operator.
So, if we raise 5 to the power of 3, we will get the
answer, 125.
So, this is the answer.
So, if I have written this basic form, a to the power of x
is equal to b.
How will we write this?
Log.
What will log be in the answer bracket?
So, log b.
And what will be the base of this?
The base here, we have a.
So, we will write that in the base.
We write in the base, which means, we write in the
subscript.
Below, below log and b, in the middle, a.
So, log b to the base a.
We will read it like this.
And what will be the answer?
The answer will be, this power.
The power of this, that is equal to x.
This is our log operator.
What is a log operator?
Its indirect meaning is, what power of a, the base, we
take, what power of the base we take, that the answer comes
b.
So, it will find out, the log operator, it will give the
answer, log operator.
That if we do x to the power of a,
If we do x to the power of a, then we will get the answer,
b.
This is what I had written.
So, basically, you got the meaning of log.
Like, if you take 2 to the power 4.
2 to the power 4, 16.
2 to 4 into 4, that is 16, you got the answer.
So, how will you write this in terms of log?
You will write, log will come here in the answer bracket.
Log 16.
And what is its base?
The base was 2.
So, what do we do to the power of 2?
That we get the answer, 16.
So, if we do 2 to the power of 4, we will get an answer, 16.
Okay? So, you understood this.
Now, see, what is the condition we have in this?
The condition is that the base,
The base a should be positive.
Take the base a as positive.
Take the base as a positive number.
And what will happen?
The value b,
The value b, you will always take as 1.
You cannot take a as 1 because the power of any number,
What will the power of 1 do to get that number?
So, no one can do the power of 1 in such a way that the
answer comes other than 1.
So, this a is not equal to 1.
What will happen?
We will not take the base as 1.
Because for 1, no matter what power you do, the answer will
always be 1.
That is why we do not take the base a as 1.
So, this is about the log operator.
In the log operator, we usually take the natural base as 10.
Or, if you study higher mathematics, then you take
exponential e.
But, that is not in our course.
So, we will not do it as such.
We will not go into that depth.
But, till now, we have taken the base as 10.
If it is not given.
Otherwise, it will be given to you.
If it is given to you to solve, then we will tell you.
That is, you take the base as 10.
Now, let us look at some examples of the log operator.
And then we will discuss its properties.
For example, you have log 10.
You have to find out log 10.
On whose base?
We take the natural base as 10.
So, what will be the answer?
What power of 10 will give us the answer as 10?
Obviously, if you take the power of 1 as 10
Then the answer will be 10.
So, the power will be your answer.
So, the power of 1 of 10 will give us the answer as 10.
So, the answer of log 10 to the base 10 is 1.
In fact, if you take the log of any number
And if its base is also the same number
Then what will be the answer?
Your answer will be always 1.
Because, if we raise any number to the power of 1
Then we will get the same number.
I will make you write this in the properties.
Suppose, I am taking the log of 1000.
1000 to the base 10.
Log 1000 to the base 10.
What will be the answer?
First, you look at 1000.
What power of 10 will give us the answer as 1000?
Indirectly, the log is asking us
What power of 10 will give us the answer as 1000? 10 to the
power 3.
So, the answer would be 3.
This was the basic questions.
We will continue with the questions and you will understand.
There are particular values of logs like you.
You can remember them.
You don't have to remember them in the questions.
You just have to see in which range they are.
These are the numbers of logs from 2 to 10.
If the number is greater than 0, then the answer should be
1.
What will be the log of 1?
If you look at the log of 2 to 10.
All the bases are 10.
So, log 10 to the base 10 is 1.
Then, the log of 9 is 0.954 which is equivalent to 0.96.
Similarly, you have all the logs from 2 to 9.
log1 is not available in this equation.
So, let us see what will be the answer of log1 to the base
10.
So, we have to find out what power of 10 we need to get 1
in the answer.
You all must be knowing that what power of any number we
need to get 1 in the answer.
So, when will we get 1 in the answer?
So, we get 1 in the answer.
So, if you make any number's power equal to 0, then you
will get 1 in the answer.
So, we have to find out the power of 10.
So, what is log1 to the base 10?
We have 0.
So, the first property you will write in the answer is log1.
This is irrespective of base.
So, irrespective of base, you will always get log1 which is
always equal to 0.
So, the first property you will write is log1 is equal to
10.
So, the second property you will get is log1 is equal to 10.
Now, I have given you an example.
If you take the same number and its same base, then when is
it possible that the number and base are same?
When the power is 1.
So, this is your second property.
Which says that if you are taking log of a number, suppose
that number is your a, and your base is also a, then it
will always be equal to 1.
Now, we have discussed this in terms of studies that what
is log and what is its meaning.
But, I can tell you a cheat code.
How can you simplify this and write log?
Suppose you have log a to the base b.
You have this.
So, you can think of this indirectly in your mind.
Log a divided by log b.
What can you think of?
There are two logs and they are getting divided.
Log a divided by log b.
In your mind, so that it becomes easier to simplify or
solve the log.
If we think in this way, then it becomes a little simple.
Now, look at the next property.
You have written 2.
Now, what will be the third property?
If you have log a and its power is n.
What is n in the power of a?
The number in the log, the argument is a to the power n.
Whatever the base is, take b, x, y, z, take anything, take
10.
So, you can keep anything.
So, what happens in this?
In the power rule.
What is the power rule in the log?
If you simplify this, then what will it be equal to?
The power will be multiplied in front of you.
What will happen?
The power will be multiplied in front of you.
And inside, what will be left?
Log a.
So, this is your third property.
That is your power rule.
So, what did we do?
We took log 0 of 1.
With the same base, if we have to take log, means the base
and the number in the log, if the argument is the same,
then our answer would be 1.
And if we have some power, then what happens to that power?
It is multiplied in front of you.
And we simply have to take log of that number.
Suppose we have to take log of 10 to the power 4.
What is the base? We are taking 10 only.
So, what will happen? 10 to the power 4.
Means log 4.
How will we solve?
There is 4 in the power.
So, 4 will be multiplied in front of you.
That would be 4 into log 10.
And we know the value of log 10.
To the base 10, that is 1.
So, this will be 1.
4 into 1, that would be 1.
We have the answer.
4.
Did you understand this rule?
Write the next rule.
The next rule that we have is your fourth rule.
If we have the terms of log inside.
What is there in the argument?
They are being multiplied.
Suppose there are two numbers in log.
What should I take in that number?
I will take a or b.
So that it is easy to remember.
You can have any particular base.
x, y, z.
So, this is log a into b.
The term inside log, what is it called?
Argument.
Whatever term is there inside log, you can call it argument.
Now, what will this be equal to?
This will be log.
The multiplication in log changes in addition.
If log is distributed, then it will change in addition.
So, log a.
Whatever is there in the base, will remain here.
Plus log b.
And what will be the base of both?
The base of both will be same.
Whatever is there in the question, will be distributed in
the base of both.
Got it?
If there is multiplication in log, then how will it open?
Log will be distributed in a and b.
In addition, what will happen in the middle?
Plus sign will be there.
Now, if you see the second rule of b.
For this product, if you see it in quotient or division.
So, if log a divided by b is there.
What is it?
Log a divided by b.
So, what will be this answer?
Simple.
It will become simple.
What will happen in log divide?
You can write it as 1 by b.
So, in divide, there is only one property.
In multiplication, there is plus sign in the middle.
In divide, there will be subtraction in the middle.
Which means, log of numerator minus log of denominator.
So, log of a minus log of...
What will be the answer?
Got it?
So, this is the corollary.
It is not a new result.
It is derived from this.
I will write it on the side.
You solve it once.
I am asking you the answer of log of 1 by b.
What will be the answer of log of 1 by b?
First, apply the quotient rule.
So, what will be the answer of log of 1 minus log of b?
What will be the answer of log of 1 minus log of b?
What is the value of log of 1?
It is 0.
So, 0 minus log b is equal to minus log b only.
You can write it as a new corollary.
Or you can solve it with the help of this.
But, if you remember it directly in the exam, it is better.
It is not a calculation of one second.
Log 1 minus log b.
Log 1 is 0.
0 minus log b is equal to minus log b.
You can do it so quickly.
I have written it here for reference.
You can do it.
Next property is the sixth property.
See the sixth property.
What is the sixth property?
Change of base.
If you want to switch the base.
Suppose you have log a to the base b.
You have log a to the base b.
You want to interchange them.
You can do it. If you want to bring A to the base but B to
the top, then simply reciprocal will be 1 upon log B to the
base A.
Now if I want to explain this to you, how can I do it?
Do you remember what I had you write?
That you can write log A to the base B as log A by log B.
I have told you that you can understand it in this way.
Now can I write it as 1 upon log B by log A?
Can I bring the numerator to the denominator of the
denominator? Yes.
Because the denominator of the denominator multiplies to
the numerator.
So this will change to the same.
Now how can you write log B by log A?
Log B to the base A.
I have written it in this way, so I can write it from the
other side also.
So you can write it as 1 upon log B to the base A.
So the base has changed.
First it was base B, now it is base A.
Whenever you will interchange the base and the argument,
you have to take the reciprocal.
Which we have taken.
This is the sixth property that we have studied.
That is log A to the base B is equal to 1 upon log B to the
base A.
I hope this will be clear to you.
If I write another property in the corollary,
I hope it is visible.
If you had just done the power rule,
What was the power rule?
You had the power of argument as n.
But let's suppose instead of the power of argument, you
have some power of base.
That means you have log A.
What is there in the base? B to the power n.
How will you solve this?
It is very simple.
You just use what I have just told you.
Rest you will get the result.
First of all, simplify it and write what I have told you to
do in your mind.
Log A divided by log B to the power n.
Think about it once.
Proof will look easy.
Formula is easy to memorize.
So we have written like this.
Now put the power rule in log B to the power n.
So B to the power n.
So n can be multiplied further.
So you will get log A divided by log B to the power n.
Now write log A divided by log B as it is.
What can you write log A divided by log B?
Log A to the base B.
Write it in backward form.
So 1 divided by n is as it is.
Now write this bracket as log A to the base B.
So this is your new type of formula.
That is log A to the base B to the power n.
If there is any power of base, then the power of base will
be multiplied in the denominator.
So 1 divided by n is log A.
Now the power of base is removed.
The power of argument was multiplied in the numerator.
And the power of base will be multiplied in the denominator.
So this can be your corollary.
You can note it along with this.
Now next formula.
You have switched the bases.
That is you have interchanged A with B.
Or you can say you have switched A and B.
Here B was the base and here B was the argument.
Here A was the argument and here B was the base.
Next is your 7th formula.
If you want to change the base.
You don't have to switch the base.
You have to change it.
Sometimes in questions, we get answers in a simpler way.
They get cancelled out.
So we can manipulate the answers in that way.
So what is the formula for that?
If you have log A to the base B.
So if you want to change the base.
You can write log A to the base C.
Suppose C is a new number.
So log A to the base C.
C is a new base.
It will be multiplied by log C to the base B.
Or you can write it in divide.
What will you write in divide?
As soon as this log goes in divide.
So what happens?
These two get interchanged.
So this will become log B to the base C.
Or you can write it as log A to the base C.
Divided by log B to the base C.
Don't write this formula.
Just keep one form in mind.
You will get the last two formulas in the same way.
Now how did this come?
How did I write this?
If you open it and write.
Log A divided by log C will come.
And what will come in multiply?
Log C divided by log B.
So this log C which came in divide.
And this log C which is in multiply.
These two will cancel out.
What will be the remaining?
Log A divided by log B.
Which is equal to log A to the base B.
Did you understand?
So this is the switching of base.
Sorry, not inter-switching.
Change of base.
We are changing the base.
Change of base.
And this sixth was switching of base.
Switching of base.
I hope all the formulas will be clear to you.
Now what we will do is.
In the next video we will see questions on the basis of
these formulas.
This video will be very long.
Otherwise I thought we will cover the questions in this
video.
But now you have a new concept of log.
So it will take some time to bring it in your mind.
So take your time.
And then revise it for 10-15 minutes.
Then you do the next video.
That is a question video.
In that we will see the remaining questions.
I hope all the formulas of log are clear to you.
Read it carefully once.
Otherwise it will happen again.
See you in the next video.
Thank you for watching the video.
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