Question

which system of equations could be graphed to solve the equation below?
$log_{0.5} x = log_{3} 2 + x$
$\bigcirc$ $y_1 = \frac{log0.5}{x}, y_2 = \frac{log3}{2+x}$
$\bigcirc$ $y_1 = \frac{log x}{log0.5}, y_2 = \frac{log2+x}{log3}$
$\bigcirc$ $y_1 = \frac{log0.5}{log3}, y_2 = \frac{log x}{log2+x}$
$\bigcirc$ $y_1 = \frac{log x}{log0.5}, y_2 = \frac{log2}{log3} + x$

Explanation:

Step1: Apply change of base rule

Recall the change of base formula: $\log_b a = \frac{\log a}{\log b}$.
For $\log_{0.5} x$, this becomes $\frac{\log x}{\log 0.5}$.
For $\log_3 2$, this becomes $\frac{\log 2}{\log 3}$.

Step2: Split the original equation

Set each side of $\log_{0.5} x = \log_3 2 + x$ equal to a separate $y$ variable:
Left side: $y_1 = \log_{0.5} x = \frac{\log x}{\log 0.5}$
Right side: $y_2 = \log_3 2 + x = \frac{\log 2}{\log 3} + x$

Step3: Match to the options

Compare the derived $y_1$ and $y_2$ to the given choices.

Answer:

D. $Y_1 = \frac{\log x}{\log 0.5}, Y_2 = \frac{\log 2}{\log 3} + x$