Today, we learn about a very interesting
concept of logarithm.
What is that? Suppose, you have log a to
the base b. We know, logs exist in such
forms. Log 4 to the base 2, okay! Now, with
the help of this, can you tell me how can we find
out the opposite, like log b to the base
a? i.e. For example, log 2 to the base
4. So, they interchange their places.
Well, this is a very easy number. You can
even tell that yourself. But suppose, it
is a, if it is a big number. You are given
this and you are asked this, just with
the help of whatever is given. Then
how will we do that. Well, there is one
formula knows as the reciprocal formula for
such situations. But how did we get that!
First we will see that.
What we will do is put m is equal to b
and let's see what happens. So, log b to
the base a is equal to log, we are
putting m as b. So, b to the base b by log a to
the base b.
How will it help us? Well, it will ease our work.
See, log b to the base a is equal to
log b to the base b. What will be the answer!
It will be 1. We know, log b to the base b or
log a to the base a, they have, if they have
the same base as a number here,
the answer is always 1. So, 1 by log a to
the base b. See, this is actually what we
wanted. Look.
Log b to the base a is actually equal to 1
by log a to the base b. See, here both have
interchanged their places. This is what we
call the reciprocal formula. See, now b to
the base a is equal to 1 by log a to the
base b. Where can you use it? Well, actually,
whenever you’ll do sums and you will
be given such logs in the denominator.
Now, such things don’t look good. It looks
good in the numerator. So, if you want to
bring it up to the numerator, you can
just interchange the places of the
base and the number given here and put
the denominator in the numerator. Or if
you want, for the ease of your sums, you
can also do the opposite.
Now tell me, log 5 to the base 10, using the
reciprocal formula, what,
how will it look?
1 by log
10 to the base 5.
Like this. So, this is how we use the
reciprocal formula.
Now, does this actually work? Like the answer is
really same or not?
let's check.
We know, log a to the base 2, what will
be the answer?
It will be
2 into 2 into 2. Two multiplied with
itself three times.
Right! Now, what if I want to find the
opposite. See, if this is a and this is b.
So, log a to the base b. Now, I’m changing
their positions. So, log b to the base a.
See, you can see I'm changing their
positions. Now, what will be the answer of
this. Tell me?
8 multiplied with itself, how many
times will give you two?
Well, 2 is a small number. So, that means
the power that will come on 8 will be a
smaller or a fraction!
8 cube rooted three times, gives you 2. Am I
correct! 2 into 2 into 2,8. So that makes 8 cube
rooted gives you 2. Or, you can say, 8 to
the power 1 by 3 gives you 2.
Right?
So, so we get log 2 the base 8 is 1 by 3!
So see, do you get the relation here?
I told you that the reciprocal formula is what?
One by this. So, log 8 to the base 2 is
3 and log 2 to the base 8 is 1 by 3.
So see you get the reciprocal of what
you were getting before! That means log
a to the base b is equal to 1 by
log b to the base a.
Right? So, this is proven.
Now see we have been given such a sum. 1
by log n to the base a plus 1 by log
n to the base c. Suppose, you go to do it the
normal fraction, that additional or fraction
way. It will become very difficult to
Solve. So, what we will do is, we will apply
the reciprocal formula to take the
denominator and make it the numerator of
here. So, do this, tell me how will you
convert this to the numerator or how will you
apply the reciprocal formula to turn
these two?
See, log n to the base a becomes log a to
the base n, plus log n to the base c
becomes log c to the base n. This is how
you convert!
Now, suppose I tell you that log a to the
base n plus log c to the base n, is
equal to 2 by log n to the base b. Can
you also apply the reciprocal formula
here?
Well, obviously, you can apply it. Tell me how
will you apply the reciprocal formula to this?
Just interchange their places.
Right!
Now, if you have log a to the base n
plus log c to the base n, is equal to 2
log b to the base n. You can see all
the bases here are now same. Prove ac
is equal to b square. Well, how will you
prove that? Prove ac is equal to b
square. Well, you just need a quick recap
of the previous three standard laws of
logarithm we had learned. Remember, the product-
law and the power-law? Well, here you have to
use that. Tell me how will you prove this?
Look at L.H.S, first.
See, log a to the base n plus log c to
the bas n. We need ac. So, actually what
you need to do is apply the product law
but in a different way. You remember the
product low? What does it state? log mn to
the base a is equal to log m to the
base a plus log n to the base a. Now, we
will apply it the opposite way. Log a to
the base n plus log c to the base n can be
written as, log ac to the base n.
Right! Now, look at the R.H.S. part.
2 log b to the base n. Remember the
power-law? Well, what do we do! We bring
the power 2 in front of the logarithm. What
we will do is, take back this power
here. This becomes log
b square to the base n. Now, you know
that log x is equal to log y. If log x
is equal to log y, then x is equal to y!
Right! So, the same thing goes here.
This is log x with the same base, is
equal to log y to the same base n. So, we can
derive that ac is equal to b square.
So, this is where we come to the end of our
solution.
So, you see how reciprocal formula helps
in performing operations in logarithm.
