Find 48 in terms of α if 36 = α. The value of 48 is ____.

Question

Find  48   in terms of α  if 36 = α. The value of 48 is ____.

Accepted Answer

We know that the basic formula of logarithm says that:

If b = c, then a^{c} = b

So, we can say that logarithm is the inverse of the exponentiation function.
Now, we are going to need to use some basic formulae of logarithm:

b \times c = b + c

This formula defines the rule if you have to split a composite exponent inside a logarithm as the sum of its constituent factors.

b = \frac{1}{a}

This formula defines the rule if I have to inverse the base and exponent of a logarithm function.
loga n=logab
This formula defines the rule when the exponent is raised to a power.
And,

a = 1

, because

a^{1} = a

Now, in this question, we are given

36

=α.
Since 62 = 36, so

6^{2}

= 2

6

(from using the above formula)
So, 2

6

= α
Also, using the above formula, we can say that:

6

=

\frac{1}{12}

Hence, α = 2×

\frac{1}{12}

or,

\frac{α}{2} = \frac{1}{12}

Now,

12

=

6

+

2

=

1 + \frac{1}{6}

Now,

6

=

2

+

3

=1+

3

Hence,

12

=

1 + \frac{1}{1 + 3}

=

\frac{1 + 3}{1 + 3}

+

\frac{1}{(1 + 3)}

=

\frac{1 + 1 + 3}{1 + 3}

=

\frac{2 + 3}{1 + 3}

(taking the LCM)
So,

\frac{α}{2} = \frac{1 + 3}{2 + 3}

And if we take the reciprocal on both sides, we get,

\frac{2}{α} = \frac{2 + 3}{1 + 3}

Now, cross-multiplying the expressions on the two sides of the equality:
2+2

3

= 2α+α

3

Rearranging the terms, we get:
2

3

−α

3

= 2α−2
Taking commons from the two sides:

3

(2−α)=2(α−1)
So,

3

=

\frac{2 (α - 1)}{2 - α}

Now,

48 = 24 + 2 = 1 + \frac{1}{24}

Now,

24 = 8 + 3 = 2^{3} + 3 = 3 + 3

Putting the value of

3

from above we have:

24 = 3 + \frac{2 (α - 1)}{2 - α}

Taking the LCM and solving, we have:

24 = \frac{6 - 3 α + 2 α - 2}{2 - α} = \frac{4 - α}{2 - α}

Substituting the value, we get:

48 = 1 + \frac{1}{24}

Now, putting the value of

24 = \frac{4 - α}{2 - α}

, we get:

48 = 1 + \frac{1}{\frac{4 - α}{2 - α}}

48 = 1 + \frac{2 - α}{4 - α} =

48 = \frac{4 - α + 2 - α}{4 - α} = \frac{6 - 2 α}{4 - α} = \frac{2 (3 - α)}{4 - α} =

Hence, the value of

48 = \frac{2 (3 - α)}{4 - α} .

Find $48$ in terms of α if $36$ = α. The value of $48$ is ____.

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