what is the solution to the equation below?\n\\(\\log_{6}4x^{2}-\\log_{6}x = 2\\)\n\\(x=\frac{1}{12})\n\\(x=\…

what is the solution to the equation below?\n\\(\\log_{6}4x^{2}-\\log_{6}x = 2\\)\n\\(x=\frac{1}{12})\n\\(x=\frac{3}{2})\n\\(x = 3)\n\\(x = 9)

what is the solution to the equation below?\n\\(\\log_{6}4x^{2}-\\log_{6}x = 2\\)\n\\(x=\frac{1}{12})\n\\(x=\frac{3}{2})\n\\(x = 3)\n\\(x = 9)

Answer

Explanation:

Step1: Use log - subtraction rule

According to the rule $\log_aM-\log_aN = \log_a\frac{M}{N}$, we can rewrite the left - hand side of the equation $\log_64x^{2}-\log_6x$ as $\log_6\frac{4x^{2}}{x}=\log_64x$. So the equation becomes $\log_64x = 2$.

Step2: Convert to exponential form

The logarithmic equation $\log_ax=b$ is equivalent to the exponential equation $a^{b}=x$. For $\log_64x = 2$, we have $6^{2}=4x$.

Step3: Solve for x

Since $6^{2}=36$, the equation $36 = 4x$. Divide both sides by 4: $x=\frac{36}{4}=9$.

Answer:

$x = 9$