Question

Решите уравнение log_2(25^х -1) = 2+log_2(5^(х+3)+1)

Answer 1

$log_2(25^{x}-1)=2+log_2(5^{x+3}+1); ;\\ ODZ:; left { {{25^{x}-1 textgreater 0} atop {5^{x+3}+1 textgreater 0}} right. ; left { {{5^{2x} textgreater 1} atop {5^{x}cdot 5^3 textgreater -1}} right. ; ; Rightarrow ; ; 2x textgreater 0; ,; x textgreater 0\\log_2(5^{2x}-1)=log_24+log_2(5^{x+3}+1)\\5^{2x}-1=4(5^{x}cdot 5^3+1)\\5^{2x}-500cdot 5^{x}-5=0\\t=5^{x} textgreater 0,; ; ; t^2-500t-5=0\\D/4=62505=9cdot 6945\\t_1=250-3sqrt{6945} textless 0\\t_2=250+3sqrt{6945} textgreater 0\\5^{x}=250+3sqrt{6945}\\x=log_5(250+3sqrt{6945})$