тренажер по теме показательные уравнения 10 класс подготовка к ЕГЭ

[latex]1.\;(0,5)^x=\frac1{64}\\\left(\frac12\right)^x=\left(\frac12\right)^6\\x=6\\\\2.\;\sqrt[3]{128}=4^{2x}\\128^{\frac13}=4^{2x}\\\left(2^7\right)^{\frac13}=(2^2)^{2x}\\2^{\frac73}=2^{4x}\\\frac73=4x\\x=\frac7{12}\\\\3.\;3^{x^2-4x-0,5}=81\sqrt3\\3^{x^2+4x-0,5}=3^4\cdot3^{\frac12}\\3^{x^2+4x-0,5}=3^{4,5}\\x^2+4x-0,5=4,5\\x^2+4x-5=0\\D=16+4\cdot5=36\\x_{1,2}=\frac{-4\pm6}2\\x_1=-5,\;x_2=1[/latex] [latex]4.\;\left(\frac37\right)^{3x-7}=\left(\frac73\right)^{7x-3}\\\left(\frac37\right)^{3x-7}=\left(\frac37\right)^{3-7x}\\3x-7=3-7x\\3x+7x = 3+7\\10x=10\\x=1\\\\5.\;\left(\frac23\right)^x\cdot\left(\frac98\right)^x=\frac{27}{64}\\\left(\frac23\cdot\frac98\right)^x=\frac{3^3}{4^3}\\\left(\frac34\right)^x=\left(\frac34\right)^3\\x=3\\\\6.\;(0,4)^{x-1}=(6,25)^{6x-5}\\(2,5)^{1-x}=(2,5^2)^{6x-5}\\(2,5)^{1-x}=(2,5)^{12x-10}\\1-x=12x-10\\-x-12x=-10-1\\-13x=-11\\x=\frac{11}{13}[/latex] [latex]7.\;3^x\cdot\left(\frac13\right)^{x-3}=\left(\frac1{27}\right)^x\\3^x\cdot3^{3-x}=3^{-3x}\\3^3=3^{-3x}\\3=-3x\\x=-1\\\\8.\;5^{2x+1}-3\cdot5^{2x-1}=550\\5\cdot5^{2x}-\frac35\cdot5^{2x}=550\\\frac{22}5\cdot5^{2x}=550\\5^{2x}=550:\frac{22}5=550\cdot\frac5{22}=125\\5^{2x}=5^3\\2x=3\\x=\frac23\\\\9.\;\sqrt{3^x}\cdot5^{\frac{x}2}=225\\3^{\frac x2}\cdot5^{\frac x2}=225\\(3\cdot5)^{\frac x2}=225\\15^{\frac x2}=15^2\\\frac x2=2\\x=4[/latex] [latex]10.\;2^{\sqrt x+2}-2^{\sqrt x+1}=12+2^{\sqrt x-1}\\2^{\sqrt x+2}-2^{\sqrt x+1}-2^{\sqrt x-1}=12\\4\cdot2^{\sqrt x}-2\cdot2^{\sqrt x}-\frac12\cdot2^{\sqrt x}=12\\\frac32\cdot2^{\sqrt x}=12\\2^{\sqrt x}=8\\2^{\sqrt x}=2^3\\\sqrt x=3\\x=9\\\\11.\;5^{2x}-2\cdot5^x-15=0\\5^x=t,\;5^{2x}=t^2,\;t\ \textgreater \ 0\\t^2-2t-15=0\\D=4+4\cdot15=64\\t_{1,2}=\frac{2\pm8}2\\t_1=5\\t_2=-3\;-\;He\;nogx.\\5^x=5\\x=1[/latex] [latex]12.\;4^{x^2+2}-9\cdot2^{x^2+2}+8=0\\2^{2x^2+4}-9\cdot2^{x^2+2}+8=0\\2^{x^2+2}=t,\;2^{2x^2+4}=t^2,\;t\ \textgreater \ 0\\t^2-9t+8=0\\D=81-32=49\\t_{1,2}=\frac{9\pm7}2\\t_1=1,\;t_2=8\\2^{x^2+2}=1\\x^2+2=0\;-\;pew.\;HET\\2^{x^2+2}=8\\x^2+2=3\\x^2=1\\x=1[/latex] [latex]13.\;4^{3x^2+x}-8=2\cdot8^{x^2+\frac x3}\\(2^2)^{3x^2+x}-8=2\cdot(2^3)^{x^2+\frac x3}\\2^{6x^2+2x}-8=2\cdot2^{3x^2+x}\\2^{6x^2+2x}-2\cdot2^{3x^2+x}-8=0\\2^{3x^2+x}=t,\;2^{6x^2+2x}=t^2,\;t\ \textgreater \ 0\\t^2-2t-8=0\\D=4+4\cdot8=36\\t_{1,2}=\frac{2\pm6}2\\t_1=4\\t_2=-3\;-\;He\;nogx.\\2^{3x^2+2}=4\\2^{3x^2+2}=2^2\\3x^2+2=2\\3x^2=0\\x=0\\\\14.\;4^{x-1}+4^x+4^{x+1}=84\\\frac14\cdot4^x+4^x+4\cdot4^x=84\\\frac{21}4\cdot4^x=84\\4^x=16\\4^x=4^2\\x=2[/latex]