Question

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

Answer 1

[tex]\log_2\dfrac{x^2z}{y^2}+\log_9x^{12}y^4[/tex]

Step-by-step explanation:

Log laws

[tex]\textsf{Product law}: \quad \log_axy=\log_ax + \log_ay[/tex]

[tex]\textsf{Quotient law}: \quad \log_a\frac{x}{y}=\log_ax - \log_ay[/tex]

[tex]\textsf{Power law}: \quad \log_ax^n=n\log_ax[/tex]

Given expression:

[tex]\log_2z+2\log_2x+4\log_9y+12\log_9x-2\log_2y[/tex]

Group terms with same log base:

[tex]\implies \log_2z+2\log_2x-2\log_2y+4\log_9y+12\log_9x[/tex]

Apply the power law:

[tex]\implies \log_2z+\log_2x^2-\log_2y^2+\log_9y^4+\log_9x^{12}[/tex]

Apply the product law:

[tex]\implies \log_2x^2z-\log_2y^2+\log_9x^{12}y^4[/tex]

Apply the quotient law:

[tex]\implies \log_2\dfrac{x^2z}{y^2}+\log_9x^{12}y^4[/tex]

Answer 2

[tex]\\ \rm\Rrightarrow log_2z+2log_2x+4log_9y+12log_9x-2log_2y[/tex]

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

So

[tex]\\ \rm\Rrightarrow log_2z+log_2x^2-log_2y^2+log_9y^4+log_9x^{12}[/tex]

[tex]\\ \rm\Rrightarrow log_2(zx^2)-log_2y^2+log_9(y^4x^{12})[/tex]

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