Question

2logx=3-2log(x+3) solve for x​

Answer 1

`\large\boxed{x=\frac{-3+\sqrt{40+10√(10)}}{2}}`

Explanation:

`2\log x=3-2\log(x+3)\\\\Domain:\ x>0\ \wedge\ x+3>0\to x>-3\\\\D:x>0\\============================\\2\log x=3-2\log(x+3)\qquad\text{add}\ 2\log(x+3)\ \text{to both sides}\\\\2\log x+2\log(x+3)=3\qquad\text{divide both sides by 2}\\\\\log x+\log(x+3)=(3)/(2)\qquad\text{use}\ \log_ab+\log_ac=\log_a(bc)\\\\\log\bigg(x(x+3)\bigg)=(3)/(2)\qquad\text{use the de}\text{finition of a logarithm}\\\\x(x+3)=10^(3)/(2)\qquad\text{use the distributive property}`

`x^2+3x=10^{1(1)/(2)}\\\\x^2+3x=10^{1+(1)/(2)}\qquad\text{use}\ a^n\cdot a^m=a^(n+m)\\\\x^2+3x=10\cdot10^(1)/(2)\qquad\text{use}\ \sqrt[n]{a}=a^(1)/(n)\\\\x^2+3x=10√(10)\qquad\text{subtract}\ 10√(10)\ \text{from both sides}\\\\x^2+3x-10√(10)=0\\\\\text{Use the quadratic formula}\\\\ax^2+bx+c=0\\\\x=(-b\pm√(b^2-4ac))/(2a)\\\\a=1,\ b=3,\ c=-10√(10)\\\\b^2-4ac=3^2-4(1)(-10√(10))=9+40√(10)\\\\x=\frac{-3\pm\sqrt{40+10√(10)}}{2(1)}=\frac{-3\pm\sqrt{40+10√(10)}}{2}\\\\x=\frac{-3-\sqrt{10+10√(10)}}{2}\\otin D`