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

`\\ \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((zx^2)/(y^2)\right)+log_9(y^4x^(12))`

Answer 2

`\log_2(x^2z)/(y^2)+\log_9x^(12)y^4`

Explanation:

Log laws

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

`\textsf{Quotient law}: \quad \log_a(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(x^2z)/(y^2)+\log_9x^(12)y^4`