Question

Demonstre a seguinte igualdade. Sendo
$a > 0$
e
$b > 1$


$log_{b}( \frac{1}{a} ) = log_{ \frac{1}{b} }(a)$

​

Answer 1

Resposta:

$\boxed{\mathtt{\log_b \frac{1}{a} = \log_{\frac{1}{b}} a, \ \forall a > 0 \ e \ b > 1}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{\log_b \left ( \frac{1}{a} \right ) =} \\\\ \mathsf{\log_b a^{- 1} =} \\\\ \mathsf{- \log_b a =}$

Passando para a base 1/b,

$\\ \displaystyle \mathsf{- \frac{\log_{\frac{1}{b}} a}{\log_{\frac{1}{b}} b} =} \\\\\\ \mathsf{\frac{\log_{\frac{1}{b}} a}{- \log_{\frac{1}{b}} b} =} \\\\\\ \mathsf{\frac{\log_{\frac{1}{b}} a}{\log_{\frac{1}{b}} b^{- 1}} =} \\\\\\ \mathsf{\frac{\log_{\frac{1}{b}} a}{\log_{\frac{1}{b}} \frac{1}{b}} =} \\\\\\ \mathsf{\frac{\log_{\frac{1}{b}} a}{1} =} \\\\ \boxed{\boxed{\mathsf{\log_{\frac{1}{b}} a}}}$