Question

Solve the following equations, having given $\log 2,\log 3$, and $\log 7$.
$\begin{array}{c}{3}^{1-x-y}=4^{-y}\\ 2^{2x-1}=3^{3y-1}\end{array}\}$

Answer 1

Product Rule: ${\log }_{a}mn={\log }_{a}m+{\log }_{a}n$

Division Rule: ${\log }_a\frac{m}{n}={\log }_{a}m-{\log }_{a}n$

Power Rule: ${\log }_a{m}^n=n{\log }_{a}m$

$3^{1-x-y}=4^{-y}$

Applying log on both sides

$(1-x-y)log3=-ylog4$

$(1-x-y)log3=-2ylog2$

$=>y=\frac{log3(1-x)}{log3-2log2}$

$2^{2x-1}=3^{3y-1}$

Applying log on both sides

$(2x-1)log2=(3y-1)log3$

Substituting $y=\frac{log3(1-x)}{log3-2log2}$

$2xlog2-log2=3log3\frac{log3(1-x)}{log3-2log2}-log3$

$x=\frac{3(log3{)}^2+3log3log2-2(log2{)}^2}{3(log3{)}^2+2log2log3-4(log2{)}^2}$

$y=\frac{-log2log3}{3(log3{)}^2+2log2log3-4(log2{)}^2}$