What is log_{3}(2x+y)-1/3 log_{3}(x-y) ?

The solution to log_{3}(2x+y)-1/3 log_{3}(x-y) is log_{3}(((2x+y)(x-y)^{2/3})/(x-y))

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simplify log_{3}(2x+y)-1/3 log_{3}(x-y)

simplify

lo g_{3} (2 x + y) - \frac{1}{3} lo g_{3} (x - y)

lo g_{3} (\frac{( 2 x + y ) ( x - y ) ^{\frac{2}{3}}}{x - y})

Solution steps

lo g_{3} (2 x + y) - \frac{1}{3} lo g_{3} (x - y)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} lo g_{3} (x - y) = lo g_{3} ((x - y)^{\frac{1}{3}})

= lo g_{3} (2 x + y) - lo g_{3} ((x - y)^{\frac{1}{3}})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

lo g_{3} (2 x + y) - lo g_{3} ((x - y)^{\frac{1}{3}}) = lo g_{3} (\frac{2 x + y}{( x - y ) ^{\frac{1}{3}}})

= lo g_{3} (\frac{2 x + y}{( x - y ) ^{\frac{1}{3}}})

Simplify

\frac{2 x + y}{( x - y ) ^{\frac{1}{3}}} : \frac{( 2 x + y ) ( x - y ) ^{\frac{2}{3}}}{x - y}

= lo g_{3} (\frac{( 2 x + y ) ( x - y ) ^{\frac{2}{3}}}{x - y})

\frac{x}{2} = \frac{x}{6} - \frac{1}{4}

2 x + 1 = \frac{5}{x + 2}

\frac{- 3}{5}

8 c^{2} - 26 c + 15

(2 x + 3) (x - 1) = 2 (x - 1)

What is log_{3}(2x+y)-1/3 log_{3}(x-y) ?
The solution to log_{3}(2x+y)-1/3 log_{3}(x-y) is log_{3}(((2x+y)(x-y)^{2/3})/(x-y))