# Question #9cda7

Oct 13, 2016

$y = 3 x - 1$

#### Explanation:

We will use the following properties of exponents and of logarithms:

• ${a}^{- x} = \frac{1}{a} ^ x$
• ${\left({a}^{x}\right)}^{y} = {a}^{x y}$
• ${a}^{x} \cdot {a}^{y} = {a}^{x + y}$
• ${\log}_{a} \left({a}^{x}\right) = x$

With those

${3}^{y} / {27}^{x} = \frac{1}{3}$

$\implies {3}^{y} / {\left({3}^{3}\right)}^{x} = {3}^{- 1}$

$\implies {3}^{y} / {3}^{3 x} = {3}^{- 1}$

$\implies {3}^{y} = {3}^{3 x} \cdot {3}^{- 1}$

$\implies {3}^{y} = {3}^{3 x - 1}$

$\implies {\log}_{3} \left({3}^{y}\right) = {\log}_{3} \left({3}^{3 x - 1}\right)$

$\therefore y = 3 x - 1$