Question

Solve the following equations:
$\begin{array}{c}{a}^{2x}.b^{3y}=m^5\\ a^{3x}.b^{2}y=m^{10}\end{array}\}$

Answer 1

$a^{2x}.b^{3y}=m^5$ ......... $\left(1\right)$

$a^{3x}.b^{2}y=m^{10}$ ......... $\left(2\right)$

Applying logarithms on both the equations gives

Eq (1) $\Rightarrow \log (a^{2x}.b^{3y})=\log \left(m^5\right)$

$\Rightarrow \log \left(a^{2x}\right)+\log \left(b^{3y}\right)=\log \left(m^5\right)\Rightarrow 2x\log a+3y\log b=5\log m$

Eq (2) $\Rightarrow \log (a^{3x}.b^{2y})={\log (m}^{10})$

$\Rightarrow {\log (a}^{3x})+\log (b^{2y})={\log (m}^{10})\Rightarrow 3x\log a+2y\log b=10\log m$

Solving both the equation gives

$x=\frac{4\log m}{\log a},y=-\frac{\log m}{\log b}$