Question

1.Sa se rezolve in multimea nr.reale ecuatia 4ˣ⁺² = 2ˣ²⁺⁵
2.Sa se rezolve in multimea nr. reale ecuatia ㏒₂(2x+5) = ㏒₂(ₓ²+3x+3)
3.Sa se rezolve in multimea nr.reale ecuatia ㏒₃(ₓ²-1)=1
4. Sa se calculeze probabilitatea ca, alegand un element n din multimea {1,2,3,4,5} acesta sa verifice inegalitatea n²≤2ⁿ

Answer 1

$1) 4^{x+2}=2^{x^{2}+5}\\ (2^{2})^{x+2} =2^{x^{2}+5}\\2^{2(x+2)} =2^{x^{2}+5}\\ 2(x+2)=x^{2} +5\\2x+4=x^{2} +5\\-x^{2} +2x+4-5=0\\-x^{2}+2x-1=0\\\Delta=b^{2} -4ac=4-4*(-1)*(-1)=4-4=0\\x_{12}=\frac{-2}{-2}=1$

$2) log_{2}(2x+5)=log_{2}(x^{2}+3x+3)\\2x+5= x^{2}+3x+3\\-x^{2}+2x-3x+5-3=0\\ -x^{2} -x+2=0|*(-1)\\x^{2} +x-2=0\\ \Delta = b^{2}-4ac=1-4*1*(-2) =1+8=9 \\x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-2-\sqrt{9}}{2}=\frac{-2-3}{2}=-\frac{5}{2} \\ x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-2+\sqrt{9}}{2}=\frac{-2-3}{2}=\frac{1}{2}$

$3) log_{3}(x^{2}-1)=1\\ log_{3}(x^{2}-1)=log_{3}3\\x^{2} -1=3\\x^{2} =3+1\\x^{2} =4\\x=\sqrt{4} \\x_{1}=-2\\x_{2}=2$