Question

Решить логарифмическое уравнение: 27*4^x+8*9^x=30*6^x

Answer 1

$27*4^{x}+8*9^{x}=30*6^{x}|:4^{x}\neq 0 \\\\27+8 *(\frac{9}{4})^{x}=30*(\frac{6}{4})^{x}\\\\8*(\frac{3}{2})^{2x}-30*(\frac{3}{2} )^{x}+27=0\\\\(\frac{3}{2})^{x}=m,m>0\\\\8m^{2}-30m+27=0\\\\D=(-30)^{2}-4*8*27=900-864=36=6^{2}\\\\m_{1}=\frac{30+6}{16}=\frac{9}{4}\\\\m_{2}=\frac{30-6}{16}=\frac{24}{16}=\frac{3}{2}$

$1)(\frac{3}{2}) ^{x}=\frac{9}{4}\\\\(\frac{3}{2} )^{x}=(\frac{3}{2})^{2}\\\\x_{1}=2\\\\2)(\frac{3}{2})^{x}=\frac{3}{2}\\\\x_{2}=1$