Question

a) 3^x+1 - 2 * 3^x-2 = 75
b) 8^x+1 = 5^x+1
c) 6^2x - 4 * 6^x - 12 =0
d) 9^x - 8 * 3^x = 9
Найти х, заранее спасибо!

Answer 1

$3^{x+1}-2cdot3^{x-2}=75\\ 3^xcdot3-2cdot3^xcdot(3)^{-2}=75\\ 3^x(3-2cdot(3)^{-2})=75\\ 3^x(3- frac{2}{9})=75\\ 3^xcdot frac{25}{9}=75\\ 3^x=75cdot frac{9}{25} \\ 3^x=27\\ 3^x=3^3\\ x=3$

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$8^{x+1}=5^{x+1} |(div (5^{x+1})}\\ frac{8^{x+1}}{5^{x+1}}= frac{5^{x+1}}{5^{x+1}}\\ ( frac{8}{5})^{ x+1}=1\\ ( frac{8}{5})^{ x+1}=( frac{8}{5})^{0}\\ x+1=0\\ x=-1$

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$6^{2x}-4cdot6^{x}-12=0\\ 6^x=t\\ t^2-4t-12=0\\ t^2-4t=12\\ t^2-4t+4=14+4\\ (t^-2)^2=16\\ sqrt{(t-2)^2}=sqrt{16}\\ |t-2|=4\\ left { {{t-2=-4} atop {t-2=4}} right. Longrightarrow left { {{t=-2} atop {t=6}} right. \\ 6^x=-2\ x=varnothing\\ 6^x=6\ x=1$

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$9^x-8cdot3^x=9\\ 3^{2x}-8cdot3^x=9\\ 3^x=t\\ t^2-8t=9\\ t^2-8t-9=0\\ D=64+36=100; sqrt {D}=10\\ t_{1/2}= frac{8pm10}{2}=4pm5\\ t_1=-1\\ t_2=9\\ 3^x=-1\\ x=varnothing\\ 3^x=9\\ 3^x=3^2\\ x=2$