Question

QF Q7.) Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.

Answer 1

`\log _9\left(\sqrt[5]{(a^6b)/(81)}\right)=(-2+\log _9\left(b\right)+6\log _9\left(a\right))/(5)`


`\log _9\left(\sqrt[5]{(a^6b)/(81)}\right) =\ \textgreater \ \log _9\left(\left((a^6b)/(81)\right)^{(1)/(5)}\right)`


`(1)/(5)\log _9\left((a^6b)/(81)\right)`


`\log _9\left((a^6b)/(81)\right)=\log _9\left(a^6b\right)-\log _9\left(81\right)`


`\log _9\left(a^6b\right)-\log _9\left(81\right)`


`\log _9\left(b\right)+\log _9\left(a^6\right)-2`


`6\log _9\left(a\right)`


`(1)/(5)\left(6\log _9\left(a\right)+\log _9\left(b\right)-2\right)`


`(1)/(5)\log _9\left(b\right)+(1)/(5)\cdot \:6\log _9\left(a\right)+(1)/(5)\left(-2\right)`


`(1)/(5)\log _9\left(b\right)+(1)/(5)\cdot \:6\log _9\dleft(a\right)-(1)/(5)\cdot \:2)`


`(1)/(5)\log _9\left(b\right)+(1)/(5)\cdot \:6\log _9\left(a\right)-(1)/(5)\cdot \:2`


`(1)/(5)\cdot \:6\log _9\left(a\right)=(6)/(5)\log _9\left(a\right)`


`(1)/(5)\log _9\left(b\right)+(6)/(5)\log _9\left(a\right)-(2)/(5)`


`(\log _9\left(b\right))/(5)+\log _9\left(a\right)(6)/(5)-(2)/(5)`


`\log _9\left(a\right)(6)/(5)\::\quad (6\log _9\left(a\right))/(5)`


`(\log _9\left(b\right))/(5)+(6\log _9\left(a\right))/(5)-(2)/(5)`


`(\log _9\left(b\right))/(5)+(6\log _9\left(a\right))/(5):\quad (\log _9\left(b\right)+6\log _9\left(a\right))/(5)`
`=\ \textgreater \ (-2+\log _9\left(b\right)+6\log _9\left(a\right))/(5)`


`\text{Took a while, but here it is. Hope it helps!}`

Answer 2

Attached is the solution.
The following log properties are used:

`log(x^n) = nlog(x)`

`log((a)/(b)) = log(a) - log(b) \\ \\ log(ab) = log(a) + log(b) \\ \\ log_9 (81) = 2 \rightarrow 9^2 = 81`