Question

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

Answer 1

Answer:

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

Hence, option 'c' is true.

Step-by-step explanation:

Given the expression

$\log _b\left(\sqrt{\frac{57}{74}}\right)$

Rewrite as

$=\log _b\left(\left(\frac{57}{74}\right)^{\frac{1}{2}}\right)$

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

$=\frac{1}{2}\log _b\left(\frac{57}{74}\right)$

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

$=\frac{1}{2}\left(\log _b\left(57\right)-\log _b\left(74\right)\right)$     ∵ $\log _b\left(\frac{57}{74}\right)=\log _b\left(57\right)-\log _b\left(74\right)$

Therefore,

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

Hence, option 'c' is true.