what is the true solution to the logarithmic equation below?\n\\(\\log_2(6x)-\\log_2(\\sqrt{x}) = 2\\)\n\\(x…
what is the true solution to the logarithmic equation below?\n\\(\\log_2(6x)-\\log_2(\\sqrt{x}) = 2\\)\n\\(x = 0\\)\n\\(x=\frac{2}{9}\\)\n\\(x=\frac{4}{9}\\)\n\\(x=\frac{2}{3}\\)
Answer
Explanation:
Step1: Use logarithm subtraction rule
According to the rule $\log_a M-\log_a N = \log_a\frac{M}{N}$, we have $\log_2\frac{6x}{\sqrt{x}}=2$. Since $\frac{6x}{\sqrt{x}} = 6\sqrt{x}$ ($x>0$), the equation becomes $\log_2(6\sqrt{x}) = 2$.
Step2: Convert to exponential form
The logarithmic - exponential conversion rule is $\log_a b=c$ implies $a^c = b$. So, $2^2=6\sqrt{x}$, which simplifies to $4 = 6\sqrt{x}$.
Step3: Solve for $\sqrt{x}$
Divide both sides of the equation $4 = 6\sqrt{x}$ by 6, we get $\sqrt{x}=\frac{4}{6}=\frac{2}{3}$.
Step4: Solve for $x$
Square both sides of the equation $\sqrt{x}=\frac{2}{3}$, we have $x = (\frac{2}{3})^2=\frac{4}{9}$.
Answer:
$x=\frac{4}{9}$