Hello everyone, welcome back to a new video.
The last video we did was on logarithm operator or
logarithm function.
Now I told you to watch that video and revise that formula
for 10 minutes and then proceed to the next video.
That is question videos.
In this video, we will see the type of questions that are
based on the same concept and properties.
I hope you have revised all the basic concepts and formulas.
So you will have more fun in answering the questions.
So let's start the questions.
Let's see what is the first question of the log.
Is it visible?
Yes, it is.
So read the first question.
If log2 is equal to 0.3010 and log3 is equal to 0.4771,
then value of log512 to the base 5 is?
I told you to revise the values of log2 and log10.
I told you that there is no requirement to revise the
values as such.
You are already given the values of log2 and log3.
If you want to revise the values, then revise the values
after the decimal.
And revise the values of log2, log5, log3, and log7.
You can revise the values of these values as well.
Because we can write the rest of the values in the multiple
of these values.
So we can get the formula of these values.
So there is no need to revise the values of log2 and log5.
If you are not given the base value, then the value of
log512 to the base 5 will be 10.
So how will you do the questions?
Think about it.
First of all, I will give you a small hint.
What will you take?
You will write 512 in the factors of the given log.
So you can prime factorize 512.
You can write it in the factors of 2 and 3.
We have learned how to do prime factorization very well.
And it is very easy also.
512.
First of all, we will divide it by 2.
2 is 4 and 10 is 5 and 12 is 6.
Then 2 is 1, 2 is 4, 16 is 8.
Then again 2 is 64.
Now this will be divided by 2.
3 is not coming.
So divide it by 2.
2 is 32.
Then 2 is 16.
You know how to write 512 in the factors of 2 and 4.
We have done it in the last video.
So 2 power 4 is 1, 2, 3, 4, 5.
And 5 power is this.
So apply the rule of indices.
If 2 to the power 4 and 5 power is added, then 2 to the
power 9.
So how can we write 512?
We can rewrite it as 2 to the power 9.
So instead of 512, write 2 to the power 9 and base is 5.
So 2 to the power 9.
Do you remember something?
In the argument, 1 power is raised.
So what do we do?
Whenever there is a power in the argument, we multiply it.
This would be 9 into log 2 to the base 5.
Now log 2 is given.
So what do we do?
Do we simply put the value of log 2 here?
Which is given in the question.
0.3010.
Do we put it?
No.
You will not put that value here.
If you do, then your answer will be wrong.
Why will it be wrong?
Because I have explained the concept of base.
Here if we have nothing as base.
There is no base of 3 or 2.
There is no base of log 3 or log 2.
Then base would be 10.
Here we have base which is given.
That is 5.
Base is given as 5.
So what do we do with 5?
We cannot put the value of log 2.
Because we will not have the same base.
This value is given at base 10.
This is given at base 5.
So how do we put it?
We cannot put it.
So how will you do it?
What do we have to do?
We have to change 5 into the form of 10.
How can we write 5 in the form of 10?
So can we write 10 as 2 into 5?
So what do we write 5 as? 5 is 10 by 10.
So 5 in terms of 10.
So what do we do in base?
We write 5.
We can also write it as log 2 divided by log a to the base
b.
I told you to write it as log 5.
Now replace 5 with 10 by 2.
So I will replace 10 by 2 in this step.
This is your 10 by 2.
I have written it in another color so that you can see the
step change.
I have done it in one step.
Now tell me.
Do you remember any other property?
Can you see any property?
You should be able to see it at this stage.
How? See this.
We have a log in the form of a divided by b.
How will we write a divided by b?
How can we write a divided by b?
Log a minus log b.
So write it like this.
Now I am not writing anything in their base.
Which means your base is 10.
So now your question is 9 into log 2 divided by log 10
minus log 2.
So if everyone's base is nothing, then consider the base to
be 10.
It will remain the same.
No problem.
Now it is very simple.
What will be the value of log 10 to the base 10? 1.
So simply replace it with 1.
I am changing it here.
So replace it with 1.
I will write it in another pen so that you can understand.
I have written 1 instead of log 10.
You have to give the value of log 2.
0.301.
Put the value and you will get the answer.
What is the answer? 9 into 0.3010 divided by 1 minus 0.3010.
Solve it.
What will be the answer? 9 times 0 is 0.
9 times 1 is 9.
0 and this is 27.
2.4 will be 2.7090.
If you subtract 1 from 1, you will get 0.3010.
What will be the answer? 9 times 9 is 9.
9 times 10 is 9.
9 times 3 is 6.
What will be the answer? 0.699.
What should you do?
If you get 4 decimals, then subtract it.
Divide it.
Cancel 0 from 0.
Divide it and you will get the answer.
You will get the answer.
It was simple.
As such, log 3 was not used anywhere.
But they have given it to confuse you.
So that you do not know that you have to use log 2 only.
You should know it immediately.
Because 2, 5, 1, 2, 2 will be there.
It will be a multiple of 2.
Was it an easy question?
Or was it a tough one?
Next question.
Solve for x.
You have to find the value of x.
You have a question.
Log 3 to the base x.
Log 9 to the base x.
Log 27 to the base x.
Log 81 to the base x.
Whose addition is equal to this?
It is equal to 10.
How will you do it?
Everyone has an equal base.
Do you remember any property?
All of them are in addition.
In addition also, you have log distributed.
When you have different log in addition.
Then how can you change them?
In multiply.
Like multiply a into b.
Log a into b.
Log a plus log b.
It can be vice versa also. So, the base will be same for
all of them, x.
And they will multiply each other.
Now, will you multiply them?
Will you multiply each other?
It will take time to multiply.
So, what you do is, there is a power of 3.
All 3 have some power.
This is the power of 3, 1.
This is the power of 3, 2.
This is the power of 3, 3.
This is the power of 3, 4.
It is 4, isn't it? 9 into 9.
So, what you do is, all the power has been added.
If all of them are multiplied, then the power has been
added.
1 plus 2 plus 3 plus 4.
That is, how much is it? 3 plus 6 is 10.
3 to the power 10.
So, log 3 to the power 10 to the base x x is equal to 10.
Okay? Do you remember one more thing about power?
That we can apply the power rule.
We have seen the power.
So, multiply 10.
I am doing it in this step.
You are intelligent, you can understand.
In this step, 10 will come in front.
So, 10 will be multiplied in front.
10 multiplied is here.
So, what will happen here?
Divide. 10 to 10 will cancel.
See, I am changing the answer in this step.
The answer is 10 to 10.
What is the result after dividing?
You get 1 on this side.
I hope you are not having any problem in this step.
I am doing it in one step.
Now, tell me.
What will happen? Now, do you remember?
You used to do this.
If you have x square.
x square is equal to 4.
So, you used to take square root on both sides.
Taking square root on both sides.
So, this is square root.
This is root.
This is x.
This is plus or minus 2.
So, now to remove this log.
What do we have to do?
We have to take anti log.
Now, what is anti log?
So, simply think of it as a concept.
What is the concept?
The concept is that
The base should have this number.
And the answer should be 3.
So, the power of x is 1.
What do we have? 3.
What have we used?
We have used the definition of log.
The power of base is in the answer.
If we take the power of base as 1.
Then the answer should be 3.
So, what is the value of x? 3.
So, x is equal to 3 is our answer.
x is equal to 3 is our solution for this question.
I hope it is clear to you.
All the results that we are using.
And it is not that particularly
In Maths, it is not at all like that.
The method by which I am walking.
The footpath by which I am walking.
You should also follow the same footpath.
The trick can always be a little different.
In the question of log,
You will do something first.
You will do something later.
But the concept should be right.
The way does not matter.
Your destination matters.
In life, the way also matters.
But in Maths.
If you are going on the right path.
If you are following the right definition, concepts and
formulas.
Then different methods can be used.
To solve the same equation.
I hope it is clear to you.
Let's move on to the next question.
This was an easy one.
Based on the concept.
See, it is the same.
This is the second question based on the same.
Same.
There is no difference in this.
The difference is in the values.
There is a difference in the values.
See, all the log terms are same.
And all the adds are same.
The base is same.
So, what should I do?
Multiplication.
You will do this quickly.
Quickly.
So, what will happen? 2 into 2 square plus 2 cube.
Right? What will this become?
This will become in your multiplier.
2 into 2 square plus 2 cube.
Into so on.
2 to the power n.
All the bases are same.
So, log to the base 2 is same.
Now, what should I do?
The power will be added.
Because the bases are same.
So, the power will be added.
2 to the power 1 plus 2 plus up to n.
Right? For now, I have not told you.
To do APGP.
Just remember this.
Or do whatever you want to do.
But, in APGP,
I will give you the proof.
How will you get the answer?
Which you have.
1 plus 2 plus 3 up to n.
These are all natural numbers.
If these natural numbers are added.
From 1 to starting.
Means, if first n natural numbers are added.
Then their direct formula is.
What is their sum? n plus 1 divided by 2.
N plus 1 divided by 2.
Their sum is equal to this.
How did this come?
I will tell you in AP.
So, this will come.
2 to the power of this.
So, this is log base 2.
Now, what happened?
This is in power.
If it is written big, then don't think that this is not in
power.
This is power.
Why is the eraser not working?
This is in the power of 2.
In the power of 2.
n plus 1 divided by 2.
Where did this come from?
This is the formula of sum of first n natural numbers.
n plus 1 divided by 2.
Whatever n is.
It is your last natural number.
Which is coming in this sequence.
So, power will come ahead.
If we apply power rule.
n plus 1 divided by 2.
Into log 2 to the base 2.
So, should I put the value of log 2?
Should I put the value of log 2?
0.3010 Should I put that?
No, we will not put that.
Because there is no need to put that.
It was in base 10.
We have asked the value of log 2.
Base is also same.
So, when the same number is in the base and argument.
Then what is the value of that?
It becomes 1.
Because it will cancel out in the divide.
Log 2 by log 2.
So, what is your answer? n plus 1 divided by 2.
n plus 1 divided by 2.
Which option is this?
Find the option.
See, hints are also given.
So, D option.
What will happen in this?
It will be correct.
Next question.
It will be a heavy question.
If we solve it, it will be solved easily.
Log k is not a tough.
Deep question.
Basic formulae.
Which can be solved by normal simplification.
Find the value of x.
We will find the value of x in this.
Which satisfies the given expression.
What is the given expression? log 2 to the base 10 plus log
4x plus 1 is equal to log x plus 2 plus 1.
How will you do it?
In this, base 10 is already given.
In this, there is no base.
So, we assume 10 only.
So, how will you do it?
Now, the problem you are facing, here you do this.
What will happen in between?
You can see log and log plus.
So, what can we write in between?
Multiply and write the same log.
And what do you do with this?
Take it to this side.
You cannot take it to minus.
You can put it here.
So, what will happen?
Log 2 to the base 10.
If it is not written, it is base 10.
Plus log 4x plus 1 minus log x plus 2 is equal to 1.
This is our equation.
Isn't it?
So, this is the equation.
These two are in plus.
So, put the formula of log a plus log b in it.
They will convert in multiply.
That is log 2 into 4x plus 1.
Both are multiplied.
2 and 4x plus 1.
They are multiplied.
You have minus log x plus 2.
As it is.
It is equal to 1.
This is the sign of minus.
If it is plus, it is multiplied.
If it is minus, it is divided.
So, take it to divide.
Log 2 4x plus 1 divided by x plus 2 is equal to 1.
Now see, if the log of any term is equal to 1, when can
this happen?
When can this happen? When the base and the number remain
the same.
Isn't it? That means the base and the log A, and here the
base is B.
When the same number A remains, then?
So, if we are doing it in the base of 10, then log 10 will
be 1, because the base is 10.
So, you can write 1 as log 10 to the base 10.
If I have not written it in the base, then it is 10.
Now, what would you do if you have the power of 2, x is
equal to the power of 2, which is 4.
If the base is the same, then the power of 2 should be the
same.
So, if the log of 2 is equal to 1, then the argument of 2
should also be the same.
What is the argument of 2? 2 into 4x plus 1 divided by x
plus 2.
What is the argument of 2? 10.
So, these two should be equal to each other.
What will you get? You will get the equation to solve.
So, solve it in x.
Simple. What should I do?
Where should I solve? I should erase this part.
Now, what should you do? Write log 2 comparing.
Then, 2 into 4x plus 1 divided by x plus 2.
It is equal to 10.
If you cross multiply, then this would be 2 into 4x, that
would be 8x.
2 into 1 plus 2 is equal to 10 into x plus 2.
Distribute this also. 10x plus 2 is equal to 8x plus 2.
Sorry, it is not plus 2. It will be plus 20. 10 into 2 is
equal to 20.
Now, solve it.
What will you get?
Log of x is equal to 8x minus 8x.
That would be 2x is equal to 2 minus 20.
That is minus 18.
2x is equal to minus 18.
What is the value of x?
You will get the value of x as minus 9.
If you get minus 9, then both the values will be equal.
Got it? Easy the question.
All the rules that we have used are not extreme.
Let us move on to the next question.
Let us see the next question.
What is the question saying?
This is not log72. This is log2.
What is the base of this? 7.
Log2 to the base 7 is equal to m.
Then we have to find out the value of this.
Log28 to the base 49.
So, which one should we convert 28 and 49 into?
The given values are 2 and 7.
Let us convert 49 into 7.
Let us convert 28 into 7 and 2.
We can write 49 as 7 square.
How can we write 28? 7 into 4.
How can we write 4? 2 square.
So, write 28 as 7 into 2 square.
What will you do now?
When I told you that if you have power in the base, then it
will be multiplied in the denominator.
So, 2 will come out.
So, it will be log7 into 2 square divided by 2 into log7.
Now, what will you do?
This is multiplied inside.
So, it will open in addition.
So, we have taken 1 by 2 as common.
What is this after taking 1 by 2 as common?
Log7 plus 2 square divided by log7.
Did everyone understand this much?
We have just applied the rule of a into b.
So, it is multiplied in plus.
Now, what will you do?
Now, distribute them.
Split the denominator.
So, it will be half log7 divided by log7 minus log2 square
upon log7.
Log7 log7 will cancel out as 1.
So, it will be half into 1 minus log2 square upon log2.
Because we have given log2 to the base 7.
So, power will come in front.
What is left?
Log2.
And I can write log7 in the base.
Log2 to the base 7.
Put its value.
You are given m.
What will happen? Put m value.
Half 1 minus 2m.
See, what is this option? 1 minus.
It is not minus.
Sorry, it is plus.
It is plus here.
So, it is plus.
Which option are you getting?
Option A.
So, what is option A?
Your correct answer.
I summed it quickly because it was very easy.
For 28 and 49, you will have to write in the form of given
base and exponent.
So, there will be a cut down in the questions.
Let us see the next question.
Next question is this and this is your last question.
Let us do the last question first.
The last question in the last slide.
You can ask any question at any time.
See this question.
If logy to the base x is 100 and logx to the base 2 is 10,
then find the value of y.
What do we have to do?
We have to write the value of y.
How will we do it?
What do you write here?
So, first write it in the form.
Can you write logy by logx?
What is this?
Logx by log2 is equal to 10.
Now, multiply both sides.
Multiply both sides.
Multiply this side by log and that side by log.
So, I will subtract 100 from the middle.
I will multiply 100 by 100.
See, logx and logx cancel.
What is left?
Logy by log2 is equal to 30,000.
How can we write this?
What is the value of logy at the base?
The value at the bottom.
Logy to the base 2 is equal to 10,000.
Now, tell me.
How can we write this?
The basic definition is that the power of 2 is 1000.
So, we should get the answer y.
This is the definition of log.
We will raise the power of 2.
We should get the answer y.
So, the power of 2 is 1000 is equal to y.
We had to find the value of y.
See, the answer is y.
We had to find the value of y.
That is option C.
So, this is the answer.
Now, this is our last question for this video.
Let's see how to solve it.
See, what is given in this? 2, now see this is not in
multiply, this is addition thing, that is log is in your
power.
What is the power of 2? Log 3, 7 to the base 3 minus 7 log
2 to the base 3 is equal to what? You have to find out.
Now see, we will see a new thing in this, we will have to
find out.
Now see, the base of both is same, log 3, log 3, isn't it?
But how do we simplify this further? How do we simplify
this further? So if here 2 is there, then log 2 to the base
3 is also same, and this is also same, then we will solve
it together.
So what we have to do in some way is, we have to bring this
2 here.
So can this 2 come here easily by itself? So this 2 can
replace 7 here by itself.
I know this, but you don't know how to do it.
So that is what I want to tell you on this question, that
how does it replace? You can write it as a property or you
can remember it in exam, that this way you can interchange
these two.
That is, we can write 2 to the power log 7 base 3, 7 to the
power log 2 to the base 3.
But how did this come? We have to see this.
What is this? How did this come? So let us see how it came.
I will tell you how this will come.
On this side, its derivation type.
If you have 7, log 2 to the power of 7 base 3, this is any
value you have, take x, take it equal to x.
Now what I will do is, I will take log on both the sides.
Like in question, you square both the sides.
If you have x equal to 2, then you will square both the
sides.
You will take square of x and square of 2.
Similarly, if you take log on both the sides, then you will
write log here also.
Taking log on both the sides, you will write.
Then you will write log here also.
Log 7 and as it is, in its power, is equal to log x.
This is done.
You have log 7.
You have written log on both the sides.
Now apply the power rule.
What will happen to the power you had?
Multiply further.
So power multiplied further.
Log 2 to the base 3 into log 7 is equal to log x.
I can write.
Now what do you do to the base 3? You do to the base 3.
Which one did you do? You did the second one.
No problem.
Let us see the second one.
I gave the example of the first one.
But solve, I am doing the second one.
You can do any one.
No issue.
So you are doing this. So the base 3 which I have removed,
I can write it in divide.
Base log into divided by log 3. I have removed the base. I
have written log 3 in divide.
Now this is in divide. So this is also in divide with this.
So I can write it below 7.
So I have done a little replacement, a little manipulation.
What am I writing?
I am writing log 3 below 7.
Now when you see these two terms, log 7 upon log 3.
How can I write this?
I have written log 7 to the base 3 like this. And we
already have log 2. Is equal to log x.
Now go backwards. Means do the reverse step. What you have
done from the top, you have done from the back.
So this log 2 is here in multiply. So can I take this to
the power? Because power is multiplied.
So power can also go up. The multiplying thing can go up to
the power.
So log 2 has gone to the power. Whose? 7. So this is log 7
to the power log 2.
And what is this base? 3. What is the base you have? 3. Is
equal to log x.
Now what will you do? Remove the log. If we remove the log
from both sides, then it will become 7 log 2 is equal to x.
Wait, we have come to the wrong thing. Means what we had to
do? We had to do the reverse.
We had to do this log 7 and in its power we had to convert
like this. 7 and in the base of 7 3.
We have to interchange this. So this will become log 7 to
the base 3.
What we will do? We will just interchange this. I have done
a little mistake.
That is I have done this log.
Instead of multiplying this, there is a small problem here.
What is the problem?
I am a Maths teacher, so I got to know. Now we had to do
manipulation.
So I took this in its power. So the question we had is
solved.
What we have to do is a small manipulation. We have to
write this backwards.
Now multiply 2 into 3 or 3 into 2. In multiplying, the
order does not matter.
The answer will be the same. So what I am doing is I am
writing log 7 first and log 2 later.
What I will do is I am writing log 7 first and log 3 into
log 2.
Now can this go in the power of 2? I had done that a little
backwards.
Now can this go in the power of 2 because it is in multiply.
So this goes in the power of 2.
Log 2 goes in the power of log 7 to the base 3 is equal to
log x.
Now taking anti-log both sides, remove the log from both
sides or compare.
The inner quantity is equal to x.
I am removing the log from both sides.
So the value of x was 7 log 2 to the base 3.
Now we have the other value of x which is 2 to the power
log 7 base 3.
So what you can do is if you have this kind of power
question, so I have told you that you can interchange the
power of this exponent and the argument in the log.
Similarly, we did it on this question.
So you can write this as x.
You got the value of x.
2 log 7 base 3.
So you can write this instead of this.
What did you write instead of this? You had x.
Instead of this you wrote 2 log 7 base 3.
Now see both are same quantity.
This is minus same quantity.
Same quantity minus same quantity.
What will the answer give us?
Zero.
So our answer is zero.
I hope you are not confused.
I was confused because I did not pay attention while doing
it.
But as I knew the final result, I did backward and verified
my mistake.
So you should know how to correct your mistake.
So I corrected it.
So if you want to write a formula for this, I will write it
for you.
If you have A log B to the base x, then you can interchange
A and B.
This will be B log A.
to the base x.
This can be equal to this.
Proof derivation, I just explained to you with the help of
that question.
So, I hope you understood the questions of log and the
formulas of the previous video.
Practice the questions further.
Log as such is not difficult.
So, you will get similar type of questions which can be
solved with the same type of formulas.
Thank you for watching the video and meet you soon in the
next video.
If you refer, you earn points that you can redeem for
discounts or even cash rewards.
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Count me in. I'll make sure to spread the word to all my
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