Question

what is log 256 to the base 2root2.

Answer 1

The question is to simplify

$log_{2 \sqrt{2} }(256)$

Formula

$log_{ {a}^{m} }(n) = \frac{1}{m} log(n) \\ \\ log_{a}( {n}^{m} ) = m log_{a}(m) \\ \\ log(m) + log(n) = log(mn)$

Solution

$log_{2 \sqrt{2} }(256) \\ \\ = log_{ {2}^{ \frac{3}{2} } }( 256) \\ \\ = \frac{2}{3} log_{2}(256) \\ \\ = \frac{2}{3} log_{2}( {2}^{8} ) \\ \\ = \frac{2 \times 8}{3} log_{2}(2) \\ \\ = \frac{16}{3}$