Question

consider the equation below.

\\\log_{4}(x+3) = \log_{2}(2+x)\\

which system of equations can represent the equation?

• \\(y_1 = \frac{\log(x+3)}{\log 4}, y_2 = \frac{\log(2+x)}{\log 2}\\)
• \\(y_1 = \frac{\log x + 3}{\log 4}, y_2 = \frac{\log 2 + x}{\log 2}\\)
• \\(y_1 = \frac{\log 4}{\log 2}, y_2 = \frac{\log(x+3)}{\log(2+x)}\\)
• \\(y_1 = \frac{\log x + 3}{4}, y_2 = \frac{\log 2 + x}{2}\\)

Explanation:

Set up the system of equations

Using the Solving Exponential Equations Graphically knowledge point
To represent a single equation of the form \(f(x) = g(x)\) as a system of equations, we set each side of the equation equal to a separate dependent variable, typically \(y_1\) and \(y_2\):

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Apply the change of base formula

Using the Change of Base Formula knowledge point

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Answer:

• (A) \(y_1 = \frac{\log(x + 3)}{\log 4} , y_2 = \frac{\log(2 + x)}{\log 2}\) (Correct answer)
• (B) \(y_1 = \frac{\log x + 3}{\log 4} , y_2 = \frac{\log 2 + x}{\log 2}\)
• (C) \(y_1 = \frac{\log 4}{\log 2} , y_2 = \frac{\log(x + 3)}{\log(2 + x)}\)
• (D) \(y_1 = \frac{\log x + 3}{4} , y_2 = \frac{\log 2 + x}{2}\)