How do you expand log_bsqrt(57/74)?

Apr 16, 2017

$\frac{1}{2} {\log}_{b} 57 - \frac{1}{2} {\log}_{b} 74$

Explanation:

There are certain rules to logratithims. You can find the complete list here, but the one that applies here is the second rule:

logb(m/n) = logb(m) – logb(n)

Using this law, we can solve $\log b \sqrt{\frac{57}{74}}$:
$\log b \frac{\sqrt{57}}{\sqrt{74}}$
$\log b \sqrt{57} - \log b \sqrt{74}$
We can stop here, but I'm going to keep going and expand it as much as I can
${\log}_{b} {57}^{.5} - {\log}_{b} {74}^{.5}$
$.5 {\log}_{b} 57 - .5 {\log}_{b} 74$
$\frac{1}{2} {\log}_{b} 57 - \frac{1}{2} {\log}_{b} 74$

That's as expanded as it gets! Good work

Apr 16, 2017

${\log}_{b} \left(\sqrt{\frac{57}{74}}\right) = \frac{1}{2} {\log}_{b} 57 - \frac{1}{2} {\log}_{b} 74$

Explanation:

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

Use the rule $\log {a}^{b} = b \log a$

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

Use the rule $\log \left(\frac{a}{b}\right) = \log a - \log b$, and don't forget that the $\frac{1}{2}$ belongs to the whole expression.

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