Question

Which equation is equivalent to 3logx+log2=log3x-log2

Answer 1

Explanation:

`3\log x+\log2=\log3x-\log2\\\\\text{Domain:}\ x>0\\\\\text{Use}\\\\\log_ab^c=c\log_ab\\\\\log_ab+\log_ac=\log_a(bc)\\\\\log_ab-\log_ac=\log_a\left((b)/(c)\right)\\===================\\\log x^3+\log2=\log\left((3x)/(2)\right)\\\\\log(2x^3)=\log\left((3)/(2)x\right)\iff2x^3=(3)/(2)x\qquad\text{subtract}\ (3)/(2)x\ \text{from both sides}\\\\2x^3-(3)/(2)x=0\qquad\text{multiply both sides by 2}\\\\4x^3-3x=0\qquad\text{use distributive property}\\\\x(4x^2-3)=0\iff x=0\ \vee\ 4x^2-3=0`

`x=0\\otin\ \text{Domain}\\\\4x^2-3=0\qquad\text{add 3 to both sides}\\\\4x^2=3\qquad\text{divide both sides by 4}\\\\x^2=(3)/(4)\Rightarrow x=\pm\sqrt{(3)/(4)}\\\\x=\pm(\sqrt3)/(\sqrt4)\\\\x=-(\sqrt3)/(2)\\otin \text{Domain}\\\\\boxed{x=(\sqrt3)/(2)}\in \text{Domain}`