Logarithms Practice Test 1

Suppose that ${\log }_{2}7$ is rational number.
∴ Let ${\log }_{2}7=\frac{\mathit{p}}{\mathit{q}}$, where p, q ∈ N.
$\Rightarrow 7=2^{\frac{\mathit{p}}{\mathit{q}}}\Rightarrow 7^{\mathit{q}}=2^{\mathit{p}}$.
Obviously (1) is false, as the L.H.S. is odd while the R.H.S. is even and the two cannot be equal.
Thus,${\log }_{2}7$ is not a rational number. Also ${\log }_{2}7$ is not an integer and so it cannot be a prime number.
Hence, ${\log }_{2}7$ is an irrational number.