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

[tex] \star) log_{2}(x) = 3 - 2 log_{2}(3) + 3 log_{2}(5) [/tex]

[tex] log_{2}(x) = log_{2}( {2}^{3} ) - log_{2}( {3}^{2} ) + log_{2}( {5}^{3} ) [/tex]

[tex] log_{2}(x) = log_{2}( \frac{ {2}^{3} }{ {3}^{2} } \times {5}^{3} ) [/tex]

[tex] log_{2}(x) = log_{2}( \frac{8}{9} \times 125) [/tex]

[tex] log_{2}(x) = log_{2}( \frac{1000}{9} ) [/tex]

[tex] = > x = \frac{1000}{9}\:\in\:\mathbb{R} [/tex]

[tex] \star) log_{5}(x) = - 1 + 3 log_{5}(2) - 2 log_{5}(3) [/tex]

[tex] log_{5}(x) = log_{5}( {5}^{ - 1} ) + log_{5}( {2}^{3} ) - log_{5}( {3}^{2} ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{1}{5} ) + log_{5}( {2}^{3} ) - log_{5}( {3}^{2} ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{ \frac{1}{5} \times {2}^{3} }{ {3}^{2} } ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{ \frac{1}{5} \times 8}{9} ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{ \frac{8}{5} }{9} ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{8}{5} \times \frac{1}{9} ) [/tex]

[tex] log_{5}(x) = log_{5}( \frac{8}{45} ) [/tex]

[tex] = > x = \frac{8}{45}\:\in\:\mathbb{R} [/tex]