# How do you express \frac { 1} { 2} \log _ { c } x + 3\log _ { c } y - 5\log _ { c } x as a single logarithm?

Apr 23, 2017

${\log}_{c} \left({y}^{3} \sqrt{{x}^{-} 9}\right)$

#### Explanation:

$n {\log}_{a} b = {\log}_{a} {b}^{n}$
${\log}_{a} b + {\log}_{a} c = {\log}_{a} b \cdot c$
${\log}_{a} b - {\log}_{a} c = {\log}_{a} \left(\frac{b}{c}\right)$

$\frac{1}{2} {\log}_{c} x + 3 {\log}_{c} y - 5 {\log}_{c} x = {\log}_{c} {x}^{\frac{1}{2}} + {\log}_{c} {y}^{3} - {\log}_{c} {x}^{5} =$

$= {\log}_{c} \left(\frac{{x}^{\frac{1}{2}} {y}^{3}}{x} ^ 5\right) = {\log}_{c} \left({x}^{- \frac{9}{2}} {y}^{3}\right) = {\log}_{c} \left({y}^{3} \sqrt{{x}^{-} 9}\right)$