Question

omar wants to use a graph to solve the equation below.$log_{6}x = log_{2}(x+4)$which system of equations should omar use?$\bigcirc$ $y_1 = \frac{log 6}{x}, y_2 = \frac{log 2}{x+4}LXB0\bigcirc$ $y_1 = \frac{log 6}{log 2}, y_2 = \frac{log x}{log(x+4)}$$\bigcirc$ $y_1 = \frac{log x}{log 6}, y_2 = \frac{log(x+4)}{log 2}$

Explanation:

Step1: Recall change of base formula

The change of base formula for logarithms is $\log_b a = \frac{\log a}{\log b}$, where $\log$ represents a common logarithm (base 10) or natural logarithm (base $e$).

Step2: Rewrite each side as a function

Set $y_1$ equal to the left-hand side of the original equation, converted using the change of base formula:
$y_1 = \log_6 x = \frac{\log x}{\log 6}$
Set $y_2$ equal to the right-hand side of the original equation, converted using the change of base formula:
$y_2 = \log_2 (x+4) = \frac{\log(x+4)}{\log 2}$

Step3: Match to the correct option

Compare the derived system to the given choices to find the match.

Answer:

$y_1 = \frac{\log x}{\log 6}, y_2 = \frac{\log(x+4)}{\log 2}$ (the fourth option)