[tex]12) {3}^{x} \times {5}^{x} = 15[/tex]
[tex] {3}^{x} \times {5}^{x} = {3}^{1} \times {5}^{1} = > x = 1[/tex]
[tex]13) \sqrt{x - 1} = \sqrt{ {x}^{2} - x - 2} \: | {( \: )}^{2} [/tex]
[tex]x - 1 \geqslant 0[/tex]
[tex] {x}^{2} - x - 2 \geqslant 0[/tex]
[tex]x - 1 = {x}^{2} - x - 2[/tex]
[tex] - {x}^{2} + x + x - 1 + 2 = 0[/tex]
[tex] - {x}^{2} + 2x + 1 = 0 \: | \times ( - 1)[/tex]
[tex] {x}^{2} - 2x - 1 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 2[/tex]
[tex]c = - 1[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 2)}^{2} - 4 \times 1 \times ( - 1)[/tex]
[tex]\Delta = 4 + 4[/tex]
[tex]\Delta = 8[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a} = \frac{ - ( - 2) \pm \sqrt{8} }{2 \times 1} = \frac{2 \pm2 \sqrt{2} }{2} [/tex]
[tex]x_{1}= \frac{2 + 2 \sqrt{2} }{2} = \frac{2(1 + \sqrt{2}) }{2} = 1 + \sqrt{2} [/tex]
[tex]x_{2}= \frac{2 - 2 \sqrt{2} }{2} = \frac{2(1 - \sqrt{2} )}{2} = 1 - \sqrt{2} [/tex]
[tex]14) \frac{ {2}^{x} }{ {3}^{x} } = \frac{3}{2} [/tex]
[tex] \frac{ {2}^{ - 1} }{ {3}^{ - 1} } = \frac{ \frac{1}{2} }{ \frac{1}{3} } = \frac{1}{2} \times 3 = \frac{3}{2} = > x = - 1[/tex]
[tex]15) log_{5}(3x + 1) = 1 + log_{5}(x - 1) [/tex]
[tex] log_{5}(3x + 1) = log_{5}(5) + log_{5}(x - 1) [/tex]
[tex] log_{5}(3x + 1) = log_{5}(5(x - 1)) [/tex]
[tex]3x + 1 = 5(x - 1)[/tex]
[tex]3x + 1 = 5x - 5[/tex]
[tex]3x - 5x = - 5 - 1[/tex]
[tex] - 2x = - 6 \: | \div ( - 2)[/tex]
[tex]x = 3[/tex]
[tex]16) \frac{1}{ {2}^{x} } = \frac{ {4}^{x} }{8} [/tex]
[tex] {2}^{x} \times {4}^{x} = 8 \times 1[/tex]
[tex] {2}^{x} \times {2}^{2x} = {2}^{3} [/tex]
[tex] {2}^{x + 2x} = {2}^{3} [/tex]
[tex] {2}^{3x} = {2}^{3} = > x = 1[/tex]
[tex]17) \sqrt{2x + 3} = x + 2 \: | {( \: )}^{2} [/tex]
[tex]2x + 3 \geqslant 0[/tex]
[tex]2x + 3 = {x}^{2} + 4x + 4[/tex]
[tex] - {x}^{2} + 2x - 4x + 3 - 4 = 0[/tex]
[tex] - {x}^{2} - 2x - 1 = 0\: | \times ( - 1)[/tex]
[tex] {x}^{2} + 2x + 1 = 0[/tex]
[tex] {(x + 1)}^{2} = 0[/tex]
[tex]x + 1 = 0 = > x = - 1[/tex]
[tex]18) \sqrt{ {x}^{2} + 2x - 3} = 2 \sqrt{3} \: | {( \: )}^{2} [/tex]
[tex]{x}^{2} + 2x - 3\geqslant0[/tex]
[tex] {x}^{2} + 2x - 3 = 12[/tex]
[tex] {x}^{2} + 2x - 3 - 12 = 0[/tex]
[tex] {x}^{2} + 2x - 15 = 0[/tex]
[tex] {x}^{2} + 5x - 3x - 15 = 0[/tex]
[tex]x(x + 5) - 3(x + 5) = 0[/tex]
[tex](x - 3)(x + 5) = 0[/tex]
[tex]x - 3 = 0 = > x = 3[/tex]
[tex]x + 5 = 0 = > x = - 5[/tex]
[tex]19) log_{2}( \sqrt{x + 1} ) = 1[/tex]
[tex] log_{2}( \sqrt{x + 1} ) = log_{2}(2) [/tex]
[tex] \sqrt{x + 1} = 2 \: | {( \: )}^{2} [/tex]
[tex]x + 1 \geqslant 0[/tex]
[tex]x + 1 = 4[/tex]
[tex]x = 4 - 1[/tex]
[tex]x = 3[/tex]
[tex]20) \sqrt[3]{ {x}^{3} + x + 1 } = x \: | {( \: )}^{3} [/tex]
[tex] {x}^{3} + x + 1 = {x}^{3} [/tex]
[tex]x + 1 = 0[/tex]
[tex]x = - 1[/tex]
