15. log√3 54 - log√3 96

Используем свойство логарифмов: $$log_a b - log_a c = log_a \frac{b}{c}$$

$$log_{\sqrt{3}} 54 - log_{\sqrt{3}} 96 = log_{\sqrt{3}} \frac{54}{96} = log_{\sqrt{3}} \frac{27}{48} = log_{\sqrt{3}} \frac{9}{16} = log_{\sqrt{3}} \frac{3^2}{16}$$

$$log_{\sqrt{3}} \frac{3^2}{16} = x$$

$$({\sqrt{3}})^x = \frac{3^2}{16}$$

$$3^{\frac{x}{2}} = \frac{3^2}{16}$$

$$\frac{x}{2} = log_3 \frac{9}{16}$$

$$x = 2 log_3 \frac{9}{16} = 2(log_3 9 - log_3 16) = 2(2 - log_3 2^4) = 2(2 - 4 log_3 2) = 4 - 8 log_3 2$$

Ответ: 4 - 8log3 2