Question

Calculați, în fiecare din cazurile de mai jos, valoarea numărului real x:
$log_{2}x = 3 - 2 log_{2}3 + 3 log_{2}5 \\ log_{5}x = - 1 + 3 log_{5}2 - 2 log_{5}3$

Answer 1

$\star) log_{2}(x) = 3 - 2 log_{2}(3) + 3 log_{2}(5)$

$log_{2}(x) = log_{2}( {2}^{3} ) - log_{2}( {3}^{2} ) + log_{2}( {5}^{3} )$

$log_{2}(x) = log_{2}( \frac{ {2}^{3} }{ {3}^{2} } \times {5}^{3} )$

$log_{2}(x) = log_{2}( \frac{8}{9} \times 125)$

$log_{2}(x) = log_{2}( \frac{1000}{9} )$

$= > x = \frac{1000}{9}\:\in\:\mathbb{R}$

$\star) log_{5}(x) = - 1 + 3 log_{5}(2) - 2 log_{5}(3)$

$log_{5}(x) = log_{5}( {5}^{ - 1} ) + log_{5}( {2}^{3} ) - log_{5}( {3}^{2} )$

$log_{5}(x) = log_{5}( \frac{1}{5} ) + log_{5}( {2}^{3} ) - log_{5}( {3}^{2} )$

$log_{5}(x) = log_{5}( \frac{ \frac{1}{5} \times {2}^{3} }{ {3}^{2} } )$

$log_{5}(x) = log_{5}( \frac{ \frac{1}{5} \times 8}{9} )$

$log_{5}(x) = log_{5}( \frac{ \frac{8}{5} }{9} )$

$log_{5}(x) = log_{5}( \frac{8}{5} \times \frac{1}{9} )$

$log_{5}(x) = log_{5}( \frac{8}{45} )$

$= > x = \frac{8}{45}\:\in\:\mathbb{R}$