### Home > PC > Chapter 4 > Lesson 4.2.3 > Problem4-106

4-106.

Solve the system below. Be sure to show all of your steps.

$\operatorname{log}_2 x − \operatorname{log}_2 y = \operatorname{log}_3 (\frac{1}{3})$
$2\operatorname{log}_4 x + \operatorname{log}_4 y = \operatorname{log}_7 49$

Simplify the first equation.

$\text{log}_2\frac{x}{y}=\text{ log}_33^{-1}$
$\text{log}_2\frac{x}{y}=-1$
$\frac{1}{2}=\frac{x}{y}$
$y=2x$

Simplify the second equation.

$\text{log}_{4}x^{2}y=\text{ log}_{7}7^2$
$\text{log}_{4}x^{2}y=2$
$4^2=\textit{x}^2\textit{y}$
$16=\textit{x}^2\textit{y}$

Substitute:

$16=2\textit{x}^3$
$8=x^3$
$x=2$
$y = 2x$
$y=2(2)$
$y=4$