Show that `log_(4)18` is an irrational number.

Explanation

`log_(4)18 = log_(4)(3^(2)+2) = 2log_(4)3+log_(4)^(2) = 2(log_(2)3)/(log_(2)4) = log_(2)3+1/2`
assume the contrary, that this number `log_(2)3` is rational number.
`rArr log_(2)3 =p/q`. Since `log_(2)3 gt 0` both numbers p and q may be regarded as natural number `rArr 3=2^(p/q) rArr 2^(p) = 3^(q)`
But this is not possible for any natural number p and q. The resulting contradiction completes the proof.