Question

1. Sa se determine daca :a)√x+1=2 si b) log in baza 3 din (2x-5)=3
2 Calculati: (log in baza 2 din 4 - log in baza 5 din 125) la putere⁻
⁶

Answer 1

$\displaystyle 1a).In ~cazul~in ~care~ambele~numere~sunt~sub~radical: \\ ~~~~~~~~~~~~~~~~~~\sqrt{x+1}=2 \\ ~~~~~~~~~~~~~~~~~~\left( \sqrt{x+1}\right)^2=2^2 \\ ~~~~~~~~~~~~~~~~~~~x+1=4 \\ ~~~~~~~~~~~~~~~~~~~x=4-1 \\ ~~~~~~~~~~~~~~~~~~~ x=3 \\ In~cazul~in~care~doar~x~este~sub~radical: \\ ~~~~~~~~~~~~~~~~~~ \sqrt{x} +1=2 \\ ~~~~~~~~~~~~~~~~~~ \sqrt{x} =2-1 \\ ~~~~~~~~~~~~~~~~~~ \sqrt{x} =1 \\ ~~~~~~~~~~~~~~~~~~x= \sqrt{1} \\ ~~~~~~~~~~~~~~~~~~x=1$
$\displaystyle b).log_3(2x-5)=3 \\ log_3(2x-5)=log_33^3 \\ 2x-5=3^3 \\ 2x-5=27 \\ 2x=27+5 \\ 2x=32 \\ x= \frac{32}{2} \\ x=16$
$\displaystyle 2).(log_24-log_5125)^{-6} =(log_22^2-log_55^3)^{-6}=(2log_22-3log_55)^{-6}= \\ =(2-3)^{-6}=(-1)^{-6}= \frac{1}{(-1)^6}= \frac{1}{1^6} = \frac{1}{1} =1$