Question

Which of the following is equivalent to logb sqrt 57/74

Answer 1

`\log _b\left(\sqrt{(57)/(74)}\right)` is equivalent to
`(1)/(2)\left(\log \:_b\left(57\right)-\log \:_b\left(74\right)\right)`.

Hence, option 'c' is true.

Explanation:

Given the expression

`\log _b\left(\sqrt{(57)/(74)}\right)`

Rewrite as

`=\log _b\left(\left((57)/(74)\right)^{(1)/(2)}\right)`

`\mathrm{Apply\:log\:rule\:}\log _a\left(x^b\right)=b\cdot \log _a\left(x\right),\:\quad \mathrm{\:assuming\:}x\:\ge \:0`

`=(1)/(2)\log _b\left((57)/(74)\right)`

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

`=(1)/(2)\left(\log _b\left(57\right)-\log _b\left(74\right)\right)` ∵
`\log _b\left((57)/(74)\right)=\log _b\left(57\right)-\log _b\left(74\right)`

Therefore,

`\log _b\left(\sqrt{(57)/(74)}\right)` is equivalent to
`(1)/(2)\left(\log \:_b\left(57\right)-\log \:_b\left(74\right)\right)`.

Hence, option 'c' is true.