[tex]b) {9}^{x} - 10 \times {3}^{x} + 9 = 0[/tex]
[tex] {( {3}^{2}) }^{x} - 10 \times {3}^{x} + 9 = 0[/tex]
[tex] { ({3}^{x} )}^{2} - 10 \times {3}^{x} + 9 = 0[/tex]
[tex] {3}^{x} = t \: \: \: ,t > 0[/tex]
[tex] {t}^{2} - 10t + 9 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 10[/tex]
[tex]c = 9[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 10)}^{2} - 4 \times 1 \times 9[/tex]
[tex]\Delta = 100 - 36[/tex]
[tex]\Delta = 64[/tex]
[tex]t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a} = \frac{ - ( - 10) \pm \sqrt{64} }{2 \times 1} = \frac{10 \pm8}{2} [/tex]
[tex]t_{1}= \frac{10 + 8}{2} = \frac{18}{2} = 9 = > {3}^{x} = 9 = > x_{1}=2[/tex]
[tex]t_{2}= \frac{10 - 8}{2} = \frac{2}{2} = 1 = > {3}^{x} = 1 = > x_{2}=0[/tex]
[tex]d)3 \times {16}^{x} + 2 \times {81}^{x} = 5 \times {36}^{x} \: | \div {16}^{x} [/tex]
[tex]3 + 2 \times \frac{ {81}^{x} }{ {16}^{x} } = 5 \times \frac{ {36}^{x} }{ {16}^{x} } [/tex]
[tex]3 + 2 \times {( \frac{81}{16}) }^{x} = 5 \times {( \frac{36}{16} )}^{x} [/tex]
[tex]3 + 2 \times {( \frac{ {9}^{2} }{ {4}^{2} } )}^{x} = 5 \times {( \frac{9}{4} )}^{x} [/tex]
[tex]3 + 2 \times{ [ {( \frac{9}{4} )}^{2} ] }^{x} = 5 \times {( \frac{9}{4}) }^{x} [/tex]
[tex]3 + 2 \times {( \frac{9}{4} )}^{2 \times x} = 5 \times {( \frac{9}{4}) }^{x} [/tex]
[tex]3 + 2 \times {( \frac{9}{4} )}^{2x} = 5 \times {( \frac{9}{4} )}^{x} [/tex]
[tex]2 \times {( \frac{9}{4}) }^{2x} - 5 \times {( \frac{9}{4} )}^{x} + 3 = 0[/tex]
[tex] {( \frac{9}{4} )}^{x} = t \: \: \: ,t > 0[/tex]
[tex]2 {t}^{2} - 5t + 3 = 0[/tex]
[tex]a = 2[/tex]
[tex]b = - 5[/tex]
[tex]c = 3[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 5)}^{2} - 4 \times 2 \times 3[/tex]
[tex]\Delta = 25 - 24[/tex]
[tex]\Delta = 1[/tex]
[tex]t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a} = \frac{ - ( - 5) \pm \sqrt{1} }{2 \times 2} = \frac{5 \pm1}{4} [/tex]
[tex]t_{1}= \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2} = > {( \frac{9}{4}) }^{x} = \frac{3}{2} [/tex]
[tex] {( \frac{ {3}^{2} }{ {2}^{2} } )}^{x} = \frac{3}{2} = > {[{(\frac{3}{2})}^{2}]}^{x} = \frac{3}{2} = > {( \frac{3}{2}) }^{2x} = {( \frac{3}{2} )}^{1} = > 2x = 1 = > x_{1} = \frac{1}{2} [/tex]
[tex]t_{2} = \frac{5 - 1}{4} = \frac{4}{4} = 1 = > {( \frac{9}{4}) }^{x} = 1 = > x_{2} = 0[/tex]
[tex]e)lg( {10}^{2x + 1} + 7 \times {10}^{x + 1}) = lg( {10}^{x + 2} - 20)[/tex]
Condiții de existență :
[tex] {10}^{2x + 1} + 7 \times {10}^{x + 1} > 0[/tex]
[tex] {10}^{x + 2} - 20 > 0[/tex]
[tex] {10}^{2x + 1} + 7 \times {10}^{x + 1} = {10}^{x + 2} - 20 [/tex]
[tex] {10}^{2x + 1} + 7 \times {10}^{x + 1} - {10}^{x + 2} + 20 = 0 \: | \div 10[/tex]
[tex] {10}^{2x} + 7 \times {10}^{x} - {10}^{x + 1} + 2 = 0[/tex]
[tex] { ({10}^{x} )}^{2} + 7 \times {10}^{x} - {10}^{x} \times 10 + 2 = 0[/tex]
[tex] {10}^{x} = t \: \: \: ,t > 0[/tex]
[tex] {t}^{2} + 7t - 10t + 2 = 0[/tex]
[tex] {t}^{2} - 3t + 2 = 0[/tex]
[tex] {t}^{2} - 2t - t + 2 = 0[/tex]
[tex]t(t - 2) - (t - 2) = 0[/tex]
[tex](t - 2)(t - 1) = 0[/tex]
[tex]t - 2 = 0 = > t_{1} = 2 = > {10}^{x} = 2 [/tex]
[tex] {10}^{x} = 2 = > x_{1} = log_{10}(2) [/tex]
[tex]t - 1 = 0 = > t_{2} = 1 = > {10}^{x} = 1 = > x_{2} = 0[/tex]
[tex]f) {lg}^{2} x + 4lgx + 3 = 0[/tex]
Condiția de existență :
[tex]x > 0[/tex]
[tex]lgx = t[/tex]
[tex] {t}^{2} + 4t + 3 = 0[/tex]
[tex] {t}^{2} + 3t + t + 3 = 0[/tex]
[tex]t(t + 3) + t + 3 = 0[/tex]
[tex](t + 3)(t + 1) = 0[/tex]
[tex]t + 3 = 0 = > t_{1} = - 3 = > lgx = - 3 = > x_{1} = {10}^{ - 3} = \frac{1}{ {10}^{3} } = \frac{1}{1000} [/tex]
[tex]t + 1 = 0 = > t_{2} = - 1 = > lgx = - 1 = > x_{2} = {10}^{ - 1} = \frac{1}{10} [/tex]
[tex]g) log_{2}(3x - 1) = log_{2}( \frac{1}{2} ) [/tex]
Condiția de existență :
[tex]3x - 1 > 0[/tex]
[tex]3x - 1 = \frac{1}{2} \: | \times 2[/tex]
[tex] 6x - 2 = 1[/tex]
[tex]6x = 1 + 2[/tex]
[tex]6x = 3 \: | \div 6[/tex]
[tex]x = \frac{3}{6} [/tex]
[tex]x = \frac{1}{2} [/tex]
