Which system of equations could be graphed to solve the equation below?

[tex]\[ \log_{0.5} x = \log_3 (2 + x) \][/tex]

A. [tex]\( y_1 = \frac{\log 0.5}{x}, \quad y_2 = \frac{\log 3}{2 + x} \)[/tex]

B. [tex]\( y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log (2 + x)}{\log 3} \)[/tex]

C. [tex]\( y_1 = \frac{\log 0.5}{\log 3}, \quad y_2 = \frac{\log x}{\log (2 + x)} \)[/tex]

D. [tex]\( y_1 = \frac{\log x}{\log 0.5}, \quad y_2 = \frac{\log 2}{\log 3} + x \)[/tex]



Answer :

To solve the equation [tex]\( \log_{0.5}(x) = \log_3(2) + x \)[/tex], we can interpret it as two functions that should intersect at the solution of the given equation.

First, we analyze the left-hand side, [tex]\( \log_{0.5}(x) \)[/tex]. Using the change of base formula for logarithms, we can rewrite this as:
[tex]\[ \log_{0.5}(x) = \frac{\log(x)}{\log(0.5)} \][/tex]
Thus, the function [tex]\( y_1 \)[/tex] is:
[tex]\[ y_1 = \frac{\log(x)}{\log(0.5)} \][/tex]

Next, we look at the right-hand side of the given equation, [tex]\( \log_3(2) + x \)[/tex]. Again, using the change of base formula, [tex]\( \log_3(2) \)[/tex] can be rewritten as:
[tex]\[ \log_3(2) = \frac{\log(2)}{\log(3)} \][/tex]
Thus, the equation simplifies to:
[tex]\[ y_2 = \frac{\log(2)}{\log(3)} + x \][/tex]

So, the functions become:
[tex]\[ y_1 = \frac{\log(x)}{\log(0.5)} \][/tex]
[tex]\[ y_2 = \frac{\log(2)}{\log(3)} + x \][/tex]

Therefore, the correct system of equations that could be graphed to solve the given equation is:

[tex]\[ y_1 = \frac{\log(x)}{\log(0.5)}, \quad y_2 = \frac{\log(2)}{\log(3)} + x \][/tex]

This matches perfectly with the choice #4 in the options provided:

[tex]\[ \text{Choice: } y_1 = \frac{\log x}{\log 0.5} \cdot y_2 = \frac{\log 2}{\log 3} + x \][/tex]

Hence, the correct choice is:
[tex]\[ \boxed{4} \][/tex]