Răspuns :
[tex]\displaystyle 1). \sqrt{ \frac{27}{-3} } + \sqrt{ \frac{125}{64} } = \sqrt{-9} + \frac{ \sqrt{125} }{ \sqrt{64} } =3i+ \frac{5 \sqrt{5} }{8} =\boxed{ \frac{24i+5 \sqrt{5} }{8}} \\ 2).log_315+log_3 \frac{8}{5}-log_38=log_3 \left(15 \cdot \frac{8}{5} \right)-log_38=log_3 \frac{120}{5}-log_38= \\ =log_324-log_38=log_3 \frac{24}{8} =log_33=\boxed{1} \\ 3). \sqrt{5} -x=4 \Rightarrow -x=4- \sqrt{5} \Rightarrow x= \boxed{\sqrt{5} -4}[/tex]
[tex]\displaystyle 4).8 \cdot 3^{x-2}+2 \cdot 3^{x-1}+3^x \cdot 23=3^x(8 \cdot 3^{-2}+2 \cdot 3^{-1}+1 \cdot 23)= \\ =3^x \left(8 \cdot \frac{1}{3^2} +2 \cdot \frac{1}{3^1} +23 \right)=3^x \left( 8 \cdot \frac{1}{9} +2 \cdot \frac{1}{3} +23 \right)= \\ =3^x \left( \frac{8}{9} + \frac{2}{3} +23 \right)=3^x \left( \frac{8}{9} + \frac{6}{9} + \frac{207}{9} \right)=3^x \cdot \frac{221}{9}= \\ = \frac{3^x \cdot 221}{9} = \frac{3^x \cdot 221}{3^2} =\boxed{3^{x-2} \cdot 221} [/tex]
[tex]\displaystyle 5).log_5(10-x)=2 \\ log_5(10-x)=log_55^2 \\ 10-x=5^2 \\ 10-x=25 \\ -x=25-10 \\ -x=15 \\ \boxed{x=-15} [/tex]
[tex]\displaystyle 7).log_5(3x+4)=2 \\ log_5(3x+4)=log_55^2 \\ 3x+4=5^2 \\ 3x+4=25 \\ 3x=25-4 \\ 3x=21 \\ \boxed{x=7} [/tex]
[tex]\displaystyle 8).log_3(x^2-2x)=log_3(2x-3) \\ x^2-2x=2x-3 \\ x^2-2x-2x=-3 \\ x^2-4x=-3 \\ x^2-4x+3=0 \\ \Delta=16-12=4 \\ x_1= \frac{4+2}{2}= \frac{6}{2} =3 \\ x_2= \frac{4-2}{2}= \frac{2}{2} =1 \\ \boxed{x=3} [/tex]
[tex]\displaystyle 9).log_7(2x+1)=2 \\ log_7(2x+1)=log_77^2 \\ 2x+1=7^2 \\ 2x+1=49 \\ 2x=49-1 \\ 2x=48 \\ \boxed{x=24 } [/tex]
[tex]\displaystyle 4).8 \cdot 3^{x-2}+2 \cdot 3^{x-1}+3^x \cdot 23=3^x(8 \cdot 3^{-2}+2 \cdot 3^{-1}+1 \cdot 23)= \\ =3^x \left(8 \cdot \frac{1}{3^2} +2 \cdot \frac{1}{3^1} +23 \right)=3^x \left( 8 \cdot \frac{1}{9} +2 \cdot \frac{1}{3} +23 \right)= \\ =3^x \left( \frac{8}{9} + \frac{2}{3} +23 \right)=3^x \left( \frac{8}{9} + \frac{6}{9} + \frac{207}{9} \right)=3^x \cdot \frac{221}{9}= \\ = \frac{3^x \cdot 221}{9} = \frac{3^x \cdot 221}{3^2} =\boxed{3^{x-2} \cdot 221} [/tex]
[tex]\displaystyle 5).log_5(10-x)=2 \\ log_5(10-x)=log_55^2 \\ 10-x=5^2 \\ 10-x=25 \\ -x=25-10 \\ -x=15 \\ \boxed{x=-15} [/tex]
[tex]\displaystyle 7).log_5(3x+4)=2 \\ log_5(3x+4)=log_55^2 \\ 3x+4=5^2 \\ 3x+4=25 \\ 3x=25-4 \\ 3x=21 \\ \boxed{x=7} [/tex]
[tex]\displaystyle 8).log_3(x^2-2x)=log_3(2x-3) \\ x^2-2x=2x-3 \\ x^2-2x-2x=-3 \\ x^2-4x=-3 \\ x^2-4x+3=0 \\ \Delta=16-12=4 \\ x_1= \frac{4+2}{2}= \frac{6}{2} =3 \\ x_2= \frac{4-2}{2}= \frac{2}{2} =1 \\ \boxed{x=3} [/tex]
[tex]\displaystyle 9).log_7(2x+1)=2 \\ log_7(2x+1)=log_77^2 \\ 2x+1=7^2 \\ 2x+1=49 \\ 2x=49-1 \\ 2x=48 \\ \boxed{x=24 } [/tex]
