Question

Write as single logarithm

Answer 1

`3.\;\;log_a \sqrt{(xz^3)/(y^3)`

`4.\;\; \log _b\left((2x^4)/(y^3)\right)`

Explanation:

For #3


`\mathrm{For\; each\; term\; apply\:log\:rule}:\quad \:a\log _c\left(b\right)=\log _c\left(b^a\right)`

`(1)/(2)\log _a\left(x\right)=\log _a\left(x^{(1)/(2)}\right)`

`(3)/(2)\log _a\left(y\right)=\log _a\left(y^{(3)/(2)}\right)`

`(3)/(2)\log _a\left(z\right)=\log _a\left(z^{(3)/(2)}\right)`

so

`(1)/(2)log_ax\:-\:(3)/(2)log_ay\:+\:(3)/(2)log_az \\\\= \log _a\left(x^{(1)/(2)}\right) + \log _a\left(y^{-(3)/(2)}\right) + \log _a\left(z^{(3)/(2)}\right)`

We know that

`x^{(1)/(2) = √(x)`

`y^-(3)/(2) = (1)/(y^(3)/(2))`

`z^{(3)/(2) } = \sqrt[3]{z}`

Take the first two terms

`\mathrm{Apply\:log\:rule}:\quad \log _c\left(a\right)-\log _c\left(b\right)=\log _c\left((a)/(b)\right)`

`(1)/(2)log_ax\:-\:(3)/(2)log_ay\: \\\\= \log _a\left(x^{(1)/(2)}\right) + \log _a\left(y^{-(3)/(2)}\right) \\\\= \log _a\left(\frac{x^{(1)/(2)}}{y^{(3)/(2)}}\right)\\\\=\log _a\left(\sqrt{(x)/(y^3)}\right)\\\\`

`=\log _a\left(\sqrt{(x)/(y^3)}\right)+\log _a\left(z^{(3)/(2)}\right)\\\\= \log _a\left(\sqrt{(x)/(y^3)}z^{(3)/(2)}\right)\\\\= log_a \sqrt{(xz^3)/(y^3)`

4.
`log_b2x\:+\:3\left(log_bx\:-\:log_by\right)`

`\mathrm{Apply\:log\:rule}:\quad \log _c\left(a\right)-\log _c\left(b\right)=\log _c\left((a)/(b)\right)`'

Term in parentheses is

`\log _b\left(x\right)-\log _b\left(y\right)\\\\`

`=\log _b\left((x)/(y)\right)`

`3\left(\log _b\left(x\right)-\log _b\left(y\right)\right)\\= 3\log _b\left((x)/(y)\right)\\= \log _b\left(\left((x)/(y)\right)^3\right) \\ because \; a\log _c\left(b\right)=\log _c\left(b^a\right))`

`\log _b\left(2x\right)+\log _b\left(\left((x)/(y)\right)^3\right)\\\\= \log _b\left(2\left((x)/(y)\right)^3x\right)\\\\= \log _b\left(2\left((x)/(y)\right)^3x\right)\\\\=\log _b\left((2x^4)/(y^3)\right)`