Question

Rewrite the following expression using the properties of loy

Answer 1

Answer:

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

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:

$\implies \log_2\dfrac{x^2z}{y^2}+\log_9x^{12}y^4$

Answer 2

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

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

So

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

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

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