Respuesta :
[tex]\bf \textit{logarithm of factors}
\\\\
log_a(xy)\implies log_a(x)+log_a(y)
\\\\\\
\textit{Logarithm of rationals}
\\\\
log_a\left( \frac{x}{y}\right)\implies log_a(x)-log_a(y)
\\\\\\
\textit{Logarithm of exponentials}
\\\\
log_a\left( x^b \right)\implies b\cdot log_a(x)\\\\
-------------------------------[/tex]
[tex]\bf log_2(z)+2log_2(x)+4log_9(y)+12log_9(x)-2log_2(y) \\\\\\ log_2(z)+log_2(x^2)+4log_9(y)+12log_9(x)-2log_2(y) \\\\\\ log_2(z\cdot x^2)-2log_2(y)+4log_9(y)+12log_9(x) \\\\\\ log_2(z\cdot x^2)-log_2(y^2)+4log_9(y)+12log_9(x) \\\\\\ log_2\left( \cfrac{zx^2}{y^2} \right)+4log_9(y)+12log_9(x) \\\\\\ log_2\left( \cfrac{zx^2}{y^2} \right)+log_9(y^4)+log_9(x^{12})\implies log_2\left( \cfrac{zx^2}{y^2} \right)+log_9\left( \cfrac{y^4}{x^{12}} \right)[/tex]
[tex]\bf log_2(z)+2log_2(x)+4log_9(y)+12log_9(x)-2log_2(y) \\\\\\ log_2(z)+log_2(x^2)+4log_9(y)+12log_9(x)-2log_2(y) \\\\\\ log_2(z\cdot x^2)-2log_2(y)+4log_9(y)+12log_9(x) \\\\\\ log_2(z\cdot x^2)-log_2(y^2)+4log_9(y)+12log_9(x) \\\\\\ log_2\left( \cfrac{zx^2}{y^2} \right)+4log_9(y)+12log_9(x) \\\\\\ log_2\left( \cfrac{zx^2}{y^2} \right)+log_9(y^4)+log_9(x^{12})\implies log_2\left( \cfrac{zx^2}{y^2} \right)+log_9\left( \cfrac{y^4}{x^{12}} \right)[/tex]
