Көрсеткіштік теңдеулер (Рустюмова 3.1.2)

Шешуі: $${5^x} \cdot 5 – {5^x} = 100$$ $${5^x}(5 – 1) = 100$$ $${5^x} \cdot 4 = 100$$ $${5^x} = 100:4$$ $${5^x} = 25$$ $${5^x} = {5^2}$$ $$x = 2$$

Шешуі: $${4 \cdot {3^x} \cdot {3^2} + 5 \cdot {3^x} – 7 \cdot {3^x} \cdot 3 = 20}$$ $${{3^x}(36 + 5 – 21) = 20}$$ $${{3^x} \cdot 20 = 20}$$ $${{3^x} = 20:20}$$ $${{3^x} = 1.}$$ $${{3^x} = {3^0}}$$ $${x = 0.}$$

Шешуі: $${{3^x} \cdot 3 – 2 \cdot {3^x} \cdot {3^{ – 2}} = 25}$$ $${{3^x}\left( {3 – 2 \cdot \frac{1}{9}} \right) = 25}$$ $${{3^x} \cdot \frac{{25}}{9} = 25}$$ $${{3^x} = 25:\frac{{25}}{9}}$$ $${{3^x} = 9}$$ $${x = 2}$$

Шешуі: $${{{10}^x} – {5^x} \cdot \frac{1}{5} \cdot {2^x} \cdot \frac{1}{4} = 950}$$ $${{{10}^x} – {{(5 \cdot 2)}^x} \cdot \frac{1}{{20}} = 950}$$ $${{{10}^x} – {{10}^x} \cdot \frac{1}{{20}} = 950}$$ $${{{10}^x}\left( {1 – \frac{1}{{20}}} \right) = 950}$$ $${{{10}^x} \cdot \frac{{19}}{{20}} = 950}$$ $${{{10}^x} = 950:\frac{{19}}{{20}}}$$ $${{{10}^x} = 1000}$$ $${{{10}^x} = {{10}^3}}$$ $${x = 3}$$

Шешуі: $${{7^x} \cdot {7^2} – \frac{1}{7} \cdot {7^x} \cdot {7^x} – 14 \cdot {7^x} \cdot \frac{1}{7} + 2 \cdot {7^x} = 48}$$ $${{7^x}(49 – 1 – 2 + 2) = 48}$$ $${{7^x} \cdot 48 = 48}$$ $${{7^x} = 48:48}$$ $${{7^x} = 1}$$ $${{7^x} = {7^0}}$$ $${x = 0}$$

Шешуі: $${2 \cdot {{16}^x} – {{16}^x} – {{16}^x} \cdot \frac{1}{{16}} = 15}$$ $${{{16}^x}\left( {2 – 1 – \frac{1}{{16}}} \right) = 15}$$ $${{{16}^x}\left( {\frac{{15}}{{16}}} \right) = 15}$$ $${{{16}^x} = 15:\frac{{15}}{{16}}}$$ $${{{16}^x} = 16}$$ $${x = 1.}$$

Шешуі: $${{2^{\sqrt x }}\left( {5 – 3 \cdot \frac{1}{2}} \right) = 56}$$ $${{2^{\sqrt x }}\left( {\frac{7}{2}} \right) = 56}$$ $${{2^{\sqrt x }} = 56:\frac{7}{2}}$$ $${{2^{\sqrt x }} = 16}$$ $${{2^{\sqrt x }} = {2^4}}$$ $${\sqrt x = 4}$$ $${x = 16}$$

Шешуі: $${{2^{x – 3}}\left( {{2^2} + {2^1} + 1} \right) = 448}$$ $${{2^{x – 3}} \cdot (7) = 448}$$ $${{2^{x – 3}} = 448:7}$$ $${{2^{x – 3}} = 64}$$ $${{2^{x – 3}} = {2^6}}$$ $${x – 3 = 6}$$ $${x = 9}$$

Шешуі: $${{2^x}\left( {2 + 3 \cdot \frac{1}{2} – 5} \right) + 6 = 0}$$ $${{2^x}\left( { – 3 + \frac{3}{2}} \right) = – 6}$$ $${{2^x}\left( { – \frac{3}{2}} \right) = – 6}$$ $${{2^x} = – 6:\left( { – \frac{3}{2}} \right)}$$ $${{2^x} = 4}$$ $${x = 2}$$

Шешуі: $${{4^{x – 2}}(4 + 11) = 15 \cdot {2^{ – 4}}}$$ $${{4^{x – 2}} \cdot 15 = 15 \cdot {2^{ – 4}}}$$ $${{2^{2x – 4}} = {2^{ – 4}}}$$ $${2x – 4 = – 4}$$ $${2x = 0}$$ $${x = 0}$$

Шешуі: $${{3^x} \cdot \frac{1}{3} \cdot {2^x} \cdot 2 + {2^x} \cdot \frac{1}{2} \cdot {3^x} = \frac{7}{{36}}}$$ $${{{(3 \cdot 2)}^x} \cdot \frac{2}{3} + {{(2 \cdot 3)}^x} \cdot \frac{1}{2} = \frac{7}{{36}}}$$ $${{6^x}\left( {\frac{2}{3} + \frac{1}{2}} \right) = \frac{7}{{36}}}$$ $${{6^x} \cdot \frac{7}{6} = \frac{7}{{36}}}$$ $${{6^x} = \frac{7}{{36}}:\frac{7}{6}}$$ $${{6^x} = \frac{1}{6}}$$ $${x = – 1.}$$

Шешуі: $${{3^{2x}} \cdot \frac{1}{{27}} – {3^{2x – 2}} + {3^{3 \cdot \frac{{2x}}{3}}} = 675}$$ $${{3^{2x}} \cdot \frac{1}{{27}} – {3^{2x}} \cdot \frac{1}{9} + {3^{2x}} = 675}$$ $${{3^{2x}}\left( {\frac{{{1}}}{{27}} – \frac{1}{9} + 1} \right) = 675}$$ $${{3^{2x}} \cdot \frac{{25}}{{27}} = 675}$$ $${{3^{2x}} = 675:\frac{{25}}{{27}}}$$ $${{3^{2x}} = 27 \cdot 27}$$ $${2x = 6}$$ $${x = 3}$$

Шешуі: $${{4^x} \cdot \frac{1}{4} + {4^x} + {4^x} \cdot 4 = 84.}$$ $${{4^x}\left( {\frac{1}{4} + 1 + 4} \right) = 84}$$ $${{4^x}\left( {\frac{{21}}{4}} \right) = 84.}$$ $${{4^x} = 84:\frac{{21}}{4}}$$ $${{4^x} = 16}$$ $${x = 2}$$

Шешуі: $${{5^x} \cdot 5 + {5^x} + {5^x} \cdot \frac{1}{5} = 155}$$ $${{5^x}\left( {5 + 1 + \frac{1}{5}} \right) = 155}$$ $${{5^x} \cdot \frac{{31}}{5} = 155}$$ $${{5^x} = 155:\frac{{31}}{5}}$$ $${{5^x} = {5^2}}$$ $${x = 2}$$

Шешуі: $${{3^{3x}} \cdot 3 – 4 \cdot {3^{3x – 3}} + {3^{3x – 2}} = 80}$$ $${{3^{3x}}\left( {3 – 4 \cdot \frac{1}{{27}} + \frac{1}{9}} \right) = 80}$$ $${{3^{3x}}\left( {{{3}} – \frac{4}{{27}} + \frac{1}{9}} \right) = 80}$$ $${{3^{3x}} \cdot \frac{{80}}{{27}} = 80}$$ $${{3^{3x}} = 80:\frac{{80}}{{27}}}$$ $${{3^{3x}} = 27}$$ $${3x = 3}$$ $${x = 1}$$

Шешуі: $${5^{x + 4}} \cdot {5^2} – {3^{x + 5}} \cdot {3^2} = 43 \cdot {5^{x + 4}} – 19 \cdot {3^{x + 5}}$$ $$19 \cdot {3^{x + 5}} – 9 \cdot {3^{x + 5}} = 43 \cdot {5^{x + 4}} – 25 \cdot {5^{x + 4}}$$ $$10 \cdot {3^{x + 4}} \cdot 3 = 18 \cdot {5^{x + 4}}$$ $$30 \cdot {3^{x + 4}} = 18 \cdot {5^{x + 4}}\quad | \div {3^{x + 4}}$$ $${30 = 18 \cdot \frac{{{5^{x + 4}}}}{{{3^{x + 4}}}}}$$ $${\frac{{30}}{{18}} = {{\left( {\frac{5}{3}} \right)}^{x + 4}}}$$ $${\frac{5}{3} = {{\left( {\frac{5}{3}} \right)}^{x + 4}}}$$ $${1 = x + 4}$$ $${x = – 3}$$

Шешуі: $${{2^{{x^2}}} \cdot \frac{1}{2} + {2^{{x^2}}} \cdot 4 = {3^{{x^2}}} \cdot \frac{1}{3} + {3^{{x^2}}}}$$ $${{2^{{x^2}}}\left( {\frac{1}{2} + 4} \right) = {3^{{x^2}}}\left( {\frac{1}{3} + 1} \right)}$$ $${2^{{x^2}}} \cdot \frac{9}{2} = {3^{{x^2}}} \cdot \frac{4}{3}\quad \left| { \div \left( {{2^{{x^2}}} \cdot \frac{4}{3}} \right)} \right.$$ $${\frac{9}{2}:\frac{4}{3} = \frac{{{3^{{x^2}}}}}{{{2^{{x^2}}}}}}$$ $${\frac{{27}}{8} = {{\left( {\frac{3}{2}} \right)}^{{x^2}}}}$$ $${{{\left( {\frac{3}{2}} \right)}^3} = {{\left( {\frac{3}{2}} \right)}^{{x^2}}}}$$ $${{x^2} = 3}$$ $${x = \pm \sqrt 3 }$$

Шешуі:$${{5^{2x}}(1 – 35) – {7^x}(1 – 35) = 0}$$ $${{5^{2x}} \cdot ( – 34) + 34 \cdot {7^x} = 0}$$ $${34 \cdot {7^x} = 34 \cdot {{25}^x}}$$ $${{7^x} = {{25}^x}\quad | \div {7^x}}$$ $${1 = {{\left( {\frac{{25}}{7}} \right)}^x}}$$ $${{{\left( {\frac{{25}}{7}} \right)}^0} = {{\left( {\frac{{25}}{7}} \right)}^x}}$$ $${x = 0}$$

Шешуі: $${{x^2} \cdot {2^x} + {2^x} – x \cdot {2^x} \cdot {2^{1,5}} = 0}$$ $${{2^x}\left( {{x^2} + 1 – x \cdot {2^{1,5}}} \right) = 0.}$$ $${{2^x} = 0.}$$ $${x \in \emptyset }$$ $${x^2} + 1 – x\sqrt {{2^3}} = 0$$ $${x^2} – 2\sqrt 2 x + 1 = 0$$ $${D = {{(2\sqrt 2 )}^2} – 4 = 4}$$ $${{x_{1,2}} = \frac{{2\sqrt 2 \pm 2}}{2} = \sqrt 2 \pm 1}$$

Шешуі: $${{3^x} \cdot {3^3} + {3^3} \cdot {3^x} \cdot \frac{1}{3} = 3}$$ $${{3^x}(27 + 9) = 3}$$ $${{3^x} \cdot 36 = 3}$$ $${{3^x} = 3:36}$$ $${3^x} = \frac{1}{{12}}$$ $$x = {\log _3}\frac{1}{{12}}$$ $${x = – {{\log }_3}12}$$ $${x = – \left( {{{\log }_3}3 + {{\log }_3}4} \right)}$$ $${x = – 1 – {{\log }_3}2}$$

Шешуі: $${{9^x} + {3^{2x – 1}} = {2^{x + 3,5}} + {2^{x + 0,5}}}$$ $${{3^{2x}} + {3^{2x}} \cdot \frac{1}{3} = {2^x}\left( {{2^{3,5}} + {2^{0,5}}} \right){\rm{. }}}$$ $${{3^{2x}}\left( {1 + \frac{1}{3}} \right) = {2^x}\left( {{2^{\frac{7}{2}}} + {2^{\frac{1}{2}}}} \right)}$$ $${{3^{2x}}\left( {\frac{4}{3}} \right) = {2^x}\left( {\sqrt {{2^7}} + \sqrt 2 } \right)}$$ $${{9^x} \cdot \frac{4}{3} = {2^x} \cdot \left( {{2^{\sqrt[3]{2}}} + \sqrt 2 } \right)}$$ $${9^x} \cdot \frac{4}{3} = {2^x} \cdot 9\sqrt 2 \quad \left| { \div \left( {{2^x} \cdot \frac{4}{3}} \right)} \right.$$ $${\frac{{{9^x}}}{{{2^x}}} = \frac{{9\sqrt 2 }}{{\frac{4}{3}}}}$$ $${{{\left( {\frac{9}{2}} \right)}^x} = \frac{{27}}{{2\sqrt 2 }}}$$ $${{{\left( {\frac{9}{2}} \right)}^x} = \frac{{{9^{\frac{3}{2}}}}}{{{2^{\frac{3}{2}}}}}}$$ $${{{\left( {\frac{9}{2}} \right)}^x} = {{\left( {\frac{9}{2}} \right)}^{\frac{3}{2}}}}$$ $${x = \frac{3}{2}}$$

Шешуі: $${{3^x} + {3^x} \cdot 3 + {3^x} \cdot 9 = {5^x} + {5^x} \cdot 5 + {5^x} \cdot 25}$$ $${{3^x}(1 + 3 + 9) = {5^x}(1 + 5 + 25)}$$ $${{3^x} \cdot 13 = {5^x} \cdot 31\quad \quad | \div \left( {{3^x} \cdot 31} \right)}$$ $${\frac{{13}}{{31}} = \frac{{{5^x}}}{{{3^x}}}}$$ $${\frac{{13}}{{31}} = {{\left( {\frac{5}{3}} \right)}^x}}$$ $${x = {{\log }_{\frac{5}{3}}}\frac{{13}}{{31}}}$$ $${x = {{\log }_{\frac{3}{5}}}\frac{{31}}{{13}}}$$

Шешуі: $${{{\left( {\frac{1}{3}} \right)}^2} \cdot {3^x} – 6 \cdot {3^{x – 4}} + 2 \cdot {3^x} \cdot \frac{1}{{{3^6}}} = 29}$$ $${{3^x}\left( {\frac{1}{9} – \frac{6}{{{3^4}}} + \frac{2}{{{3^6}}}} \right) = 29}$$ $${{3^x}\left( {\frac{{81 – 54 + 2}}{{729}}} \right) = 29}$$ $${{3^x} \cdot \frac{{29}}{{729}} = 29}$$ $${{3^x} = 29:\frac{{29}}{{729}}}$$ $${{3^x} = 729}$$ $${x = 6}$$

Шешуі: $${{3^{x – 2}} + {3^{x – 3}} = 99 + \sqrt {{3^{ – 8 + 2x}}} }$$ $${{3^{x – 2}} + {3^{x – 3}} – {3^{\frac{{ – 8 + 2x}}{2}}} = 99}$$ $${{3^{x – 2}} + {3^{x – 3}} + {3^{x – 4}} = 99}$$ $${{3^{x – 4}}\left( {{3^2} + {3^1} + 1} \right) = 99}$$ $${{3^{x – 4}} \cdot 11 = 99}$$ $${{3^{x – 4}} = 9}$$ $${x – 4 = 2}$$ $$x = 6.$$