Solve the following equations.
$l o g_{a^{3}} x - l o g_{a^{4}} x + l o g_{a} x = 0.75.$

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Solution

We know that ${log}_{m^{k}} n = \frac{1}{k} {log}_{m} n$

${log}_{a^{3}} x - {log}_{a^{4}} x + {log}_{a} x = 0.75$

$⟹ \frac{1}{3} {log}_{a} x - \frac{1}{4} {log}_{a} x + {log}_{a} x = \frac{3}{4}$

$⟹ \frac{13}{12} l o g_{a} x = \frac{3}{4}$

$⟹ {log}_{a} x = \frac{9}{13}$

$⟹ x = a^{9 / 13}$

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