${\log _2}7$ એ . . . . થાય.
IIT 1990, Easy
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(c) Suppose, if possible, \({\log _2}7\) is rational, say \(p/q\) where \(p\) and \(q\) are integers, prime to each other.
(c) Suppose, if possible, \({\log _2}7\) is rational, say \(p/q\) where \(p\) and \(q\) are integers, prime to each other.
Then, \({p \over q} = {\log _2}7\,\,\,\,\, \Rightarrow 7 = {2^{p/q}}\,\,\, \Rightarrow {2^p} = {7^q}\),
which is false since \(L.H.S\) is even and \(R.H.S\) is odd. Obviously \({\log _2}7\) is not an integer and hence not a prime number.
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