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Question

The number log27 is _____


A

a natural number

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B

a rational number

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C

an irrational number

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D

a negative integer

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Solution

The correct option is C

an irrational number


Suppose log27 is a rational number; say pq where p and q are integer prime to each other.

Then, pq= log27

7=2p/q

7q = 2p

The integral powers of an odd number are all odd (72=49,73=343) and the integral powers of an even number are all even (42=16,43=64).

Since RHS is even number and LHS is odd. Both are not equal. So, our assumption was false. log27 cannot be expressed in the form of pq where p and q are integers or prime.

So,log27 is an irrational number.


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