Show that there is no rational number r satisyfying 4r 7Sol

Show that there is no rational number r satisyfying 4^r = 7.

Solution

Assume that there is such a rational number r = m / n for m,n integers (n positive). Then we have
4^(m/n) = 7
take power of n on both the sides
=> [4^(m/n)]^n = 7^n
4^m = 7^n ---------(1)

It can be seen that both sides are integers. Consider the prime factorisation of both sides;
the left consists of m 4\'s, while the right consist of n 7\'s.

As , the prime factorisations are different, the equality in equation (1) cannot exist for two integers.
Hence we have a contradiction.
therefore, there is no rational number r satisyfying 4^r = 7