Question

4^x+1 - 7*10^x + 3*25^x=0
Va rog!

Answer 1

   
[tex]4^{x+1}-7\cdot 10^x +3\cdot 25^x = 0 \\ 4^x \cdot 4-7\cdot 10^x +3\cdot 25^x = 0 \\ 4\cdot 2^{2x} - 7\cdot (2\cdot 5)^x +3\cdot 5^{2x}=0 \\ 4\cdot (2^{x})^2 - 7\cdot 2^x \cdot 5^x +3\cdot (5^{x})^2=0 \\ Notatii: \\ a=2^x \\ b = 5^x \\ \\ Rescriem ~ecuatia: \\ 4a^2 - 7ab +3b^2 = 0 \\ 4a^2 - 4ab-3ab+3b^2 = 0 \\ 4a(a-b) -3b(a-b)=0 (a-b)(4a-3b)=0 \\ \\ a-b = 0~~\Rightarrow~~ a=b ~~\Rightarrow ~~2^x=5^x ~~\Rightarrow ~~\boxed{x_{_1}=0 } [/tex]


[tex]4a-3b~~\Rightarrow~4a=3b \\ 4\cdot 2^x = 3\cdot 5^x \\ \displaystyle \\ \frac{2^x}{5^x}= \frac{3}{4} \\ \\ \Big(\frac{2}{5}\Big)^x= \frac{3}{4} \\ \\ x_{_2}= \boxed{log_{ \frac{2}{5} }\frac{3}{4} } ~~~ \texttt{(Logaritm in baza 2/3 din 3/4)}[/tex]