Q:

Rewrite the following expression using the properties of logarithms. log2z+2log2x+4log9y+12log9xβˆ’2log2y

Accepted Solution

A:
[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]