Q:

Solving Equations using logarithm

Here is a system of equations for which I am having difficulty solving:
\begin{cases} a^{2x}.b^{3y}=m^5 \\ a^{3x}.b^{2y}=m^{10} \end{cases}

A:

$a^{2x}.b^{3y} = m^5$
$a^{3x}.b^{2y} = m^{10}$
Taking natural log on both sides of both equations,
$ln a^{2x} + ln b^{3y} = ln m^5$
$ln a^{3x} + ln b^{2y} = ln m^{10}$

$2xln(a) + 3yln(b) = 5ln(m)$ ---{i)
$3xln(a) + 2yln(b) = 10ln(m)$ ---(ii)

$x = \frac{2.5ln(m) - 3.10ln(m)}{2ln(a).2ln(b) - 3ln(a).3ln(b)} = \frac{-20ln(m)}{-5ln(a)ln(b)} = 4ln(m - a -b) $
$y = \frac{2.10ln(m) - 3.5ln(m)}{2ln(a).2ln(b) - 3ln(a).3ln(b)} = \frac{5ln(m)}{-5ln(a)ln(b)} = -ln(m -a - b)$

