Question

log in baza 5 din(3la x - 5la(2-x))= 3log in baza 5 din 2 + 2-x

Answer 1

$\it log_5(3^x-5^{2-x})=3log_52+2-x \Rightarrow log_5(3^x-5^{2-x})-log_5 8=2-x \Rightarrow \\ \\ \\ \Rightarrow log_5 \dfrac{3^x-5^{2-x}}{8}=2-x \Rightarrow \dfrac{3^x-5^{2-x}}{8}=5^{2-x} \Rightarrow 3^x-5^{2-x}=8\cdot5^{2-x}\Rightarrow\\ \\ \\ \Rightarrow 3^x=9\cdot5^{2-x} \Rightarrow 3^x=3^2\cdot5^{2-x}|_{:3^2} \Rightarrow 3^{x-2}=5^{2-x} \Rightarrow 3^{x-2}=\dfrac{1}{5^{x-2}} \Rightarrow \\ \\ \\ \Rightarrow 3^{x-2}\cdot5^{x-2}=1 \Rightarrow 15^{x-2}=1 \Rightarrow x-2=0 \Rightarrow x=2$