What is 2log_{3}(x)-5log_{3}(x-3)+log_{3}(y) ?

The solution to 2log_{3}(x)-5log_{3}(x-3)+log_{3}(y) is log_{3}((x^2y)/((x-3)^5))

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simplify 2log_{3}(x)-5log_{3}(x-3)+log_{3}(y)

simplify

2 lo g_{3} (x) - 5 lo g_{3} (x - 3) + lo g_{3} (y)

lo g_{3} (\frac{x ^{2} y}{( x - 3 ) ^{5}})

Solution steps

2 lo g_{3} (x) - 5 lo g_{3} (x - 3) + lo g_{3} (y)

Apply log rule:

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

2 lo g_{3} (x) = lo g_{3} (x^{2})

= lo g_{3} (x^{2}) - 5 lo g_{3} (x - 3) + lo g_{3} (y)

Apply log rule:

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

5 lo g_{3} (x - 3) = lo g_{3} ((x - 3)^{5})

= lo g_{3} (x^{2}) - lo g_{3} ((x - 3)^{5}) + lo g_{3} (y)

Apply log rule:

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

lo g_{3} (x^{2}) - lo g_{3} ((x - 3)^{5}) = lo g_{3} (\frac{x ^{2}}{( x - 3 ) ^{5}})

= lo g_{3} (\frac{x ^{2}}{( x - 3 ) ^{5}}) + lo g_{3} (y)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{3} (\frac{x ^{2}}{( x - 3 ) ^{5}}) + lo g_{3} (y) = lo g_{3} (\frac{x ^{2}}{( x - 3 ) ^{5}} y)

= lo g_{3} (\frac{x ^{2}}{( x - 3 ) ^{5}} y)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= lo g_{3} (\frac{x ^{2} y}{( x - 3 ) ^{5}})

ln (\frac{2 x}{y ^{10}})

lo g_{8} (\frac{x ^{2}}{y})

\frac{7}{4} \cdot ln (\frac{7}{4}) - \frac{1}{\frac{7}{4}}

- ln (4) + ln (\frac{1}{4})

ln (20) - ln (9)

What is 2log_{3}(x)-5log_{3}(x-3)+log_{3}(y) ?
The solution to 2log_{3}(x)-5log_{3}(x-3)+log_{3}(y) is log_{3}((x^2y)/((x-3)^5))