Find x if $5^{2 x - 1} - 25^{x - 1} = 2500.$

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Solution

$5^{2 x - 1} - 25^{x - 1} = 2500$
$\frac{5^{2 x}}{5} - 5^{2 (x - 1)} = 2500$
$\frac{5^{2 x}}{5} - \frac{5^{2 x}}{25} = 2500$
$5^{2 x} [\frac{1}{5} - \frac{1}{25}] = 2500$
$5^{2 x} [\frac{4}{25}] = 2500$
$5^{2 x} = \frac{2500 \times 25}{4}$
$5^{2 x} = 15625$
$5^{2 x} = 5^{6}$
$2 x = 6$
$x = 3$

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