How do you use the properties of logarithms to expand log_3(sqrt(a-1)/9)?

Feb 8, 2017

$\frac{1}{2} {\log}_{3} \left(a - 1\right) - 2$

Explanation:

Use the following laws of logarithms to solve this problem

$\log {a}^{n} = n \log a$
$\log \left(\frac{a}{b}\right) = \log a - \log b$
•log_an =logn/loga
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$= {\log}_{3} \sqrt{a - 1} - {\log}_{3} 9$

$= {\log}_{3} {\left(a - 1\right)}^{\frac{1}{2}} - {\log}_{3} 9$

$= \frac{1}{2} {\log}_{3} \left(a - 1\right) - \log \frac{9}{\log} 3$

$= \frac{1}{2} {\log}_{3} \left(a - 1\right) - \log {3}^{2} / \log 3$

$= \frac{1}{2} {\log}_{3} \left(a - 1\right) - \frac{2 \log 3}{\log} 3$

$= \frac{1}{2} {\log}_{3} \left(a - 1\right) - 2$

Hopefully this helps!