0 like 0 dislike
Rewrite the following expression using the properties of loy
log2 z + 2 log2 x+ 4 log9 y + 12 log9 x - 2 log2 y

0 like 0 dislike
$$\log_2\dfrac{x^2z}{y^2}+\log_9x^{12}y^4$$

Step-by-step explanation:

Log laws

$$\textsf{Product law}: \quad \log_axy=\log_ax + \log_ay$$

$$\textsf{Quotient law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay$$

$$\textsf{Power law}: \quad \log_ax^n=n\log_ax$$

Given expression:

$$\log_2z+2\log_2x+4\log_9y+12\log_9x-2\log_2y$$

Group terms with same log base:

$$\implies \log_2z+2\log_2x-2\log_2y+4\log_9y+12\log_9x$$

Apply the power law:

$$\implies \log_2z+\log_2x^2-\log_2y^2+\log_9y^4+\log_9x^{12}$$

Apply the product law:

$$\implies \log_2x^2z-\log_2y^2+\log_9x^{12}y^4$$

Apply the quotient law:

$$\implies \log_2\dfrac{x^2z}{y^2}+\log_9x^{12}y^4$$
by
0 like 0 dislike
$$\\ \rm\Rrightarrow log_2z+2log_2x+4log_9y+12log_9x-2log_2y$$

- log_a^b=bloga
- log_a(bc)=log_a b+log_a c

So

$$\\ \rm\Rrightarrow log_2z+log_2x^2-log_2y^2+log_9y^4+log_9x^{12}$$

$$\\ \rm\Rrightarrow log_2(zx^2)-log_2y^2+log_9(y^4x^{12})$$

$$\\ \rm\Rrightarrow log_2\left(\dfrac{zx^2}{y^2}\right)+log_9(y^4x^{12})$$
by