Question

Mostre que:

log2 (1/2) = log1/2 (2)

Answer 1

Olá,

Queremos mostrar que:

$\sf \: log_{2}( \frac{1}{2} ) = log_{ \frac{1}{2} }(2)$

Vamos fazer:

$\sf \: log_{2}( \frac{1}{2} ) = x \\ \sf \: e \: log_{ \frac{1}{2} }(2) = y$

Então, temos que provar que:

$\sf \: x = y$

Assim:

$\sf \: log_{2}( \frac{1}{2} ) = x < = > {2}^{x} = \frac{1}{2} \\ \sf \: {2}^{x} = {2}^{ - 1} = > \boxed{ \sf \:x = - 1 }$

Assim como:

$\sf \: log_{ \frac{1}{2} }(2) = y < = > { (\frac{1}{2} )}^{y} = 2 \\ \sf \: ( {2}^{ - 1} {)}^{y} = 2 = > - y = 2 = > \boxed{ \sf \: y = - 1}$

Desta forma, temos que:

$\sf \: x = y \: = > \boxed{ \sf \: log_{2}( \frac{1}{2} ) = log_{ \frac{1}{2} }( 2) }$