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Solving for unknowns

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Eli
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#1

Solve for $x$ and $y$ for which $(0.2)^{x}(0.5)^{y} = 200$.

Solution

$(0.2)^{x}(0.5)^{y} = 200$

$\left(\frac{2}{10} \right)^{x} \left(\frac{5}{10} \right)^{y} = (2^{3})(5^{2})$

$\left(\frac{1}{5} \right)^{x} \left(\frac{1}{2} \right)^{y} = (2^{3})(5^{2})$

$(5^{-x})(2^{-y}) = (2^{3})(5^{2})$

$(5)^{-x}(2)^{-y} = (5^{2}) (2^{3})$

Comparing powers of like terms for both sides:

$-x = 2$

$x = -2;$

$-y = 3$

$y = -3.$

Therefore $x = -2 \ \text{and} \ y = -3$.

Any Alternative approach?
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