Question

if a logarithm of a number to the base 3 is 5 then the number will be​

Answer 1

Answer:

1 < Log(base 3)5 < 2

So, log(base 5)(2+x) should also lie between 1 and 2

1 < log(base 5)(2+x) < 2

5^1 < 2+x < 5^2

5 < 2+x < 25

3 < x < 23

Option D is the right answer.

Answer 2

Answer:

Proof by contradiction:

Let log 5 at base 3 is rational, say a/b where

a and b are positive integers(check that a/b has to be positive).

So

Therefore 3 must divide 5.

But 3 and 5 are co-prime.

So, our assumption is wrong.

Hence it is irrational.