What is 1/2 log_{3}(x-3)+6log_{3}(y) ?

The solution to 1/2 log_{3}(x-3)+6log_{3}(y) is log_{3}((x-3)^{1/2}y^6)

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simplify 1/2 log_{3}(x-3)+6log_{3}(y)

simplify

\frac{1}{2} lo g_{3} (x - 3) + 6 lo g_{3} (y)

lo g_{3} ((x - 3)^{\frac{1}{2}} y^{6})

Solution steps

\frac{1}{2} lo g_{3} (x - 3) + 6 lo g_{3} (y)

Apply log rule:

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

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

= lo g_{3} ((x - 3)^{\frac{1}{2}}) + 6 lo g_{3} (y)

Apply log rule:

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

6 lo g_{3} (y) = lo g_{3} (y^{6})

= lo g_{3} ((x - 3)^{\frac{1}{2}}) + lo g_{3} (y^{6})

Apply log rule:

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

lo g_{3} ((x - 3)^{\frac{1}{2}}) + lo g_{3} (y^{6}) = lo g_{3} (y^{6} (x - 3)^{\frac{1}{2}})

= lo g_{3} ((x - 3)^{\frac{1}{2}} y^{6})

lo g_{3} (4) + lo g_{3} (7)

lo g_{10} (\frac{x ^{2} y ^{4}}{z ^{3}})

ln (\frac{53}{7})

ln (11) - 3 ln (2)

lo g_{10} (x^{4} y^{5})

What is 1/2 log_{3}(x-3)+6log_{3}(y) ?
The solution to 1/2 log_{3}(x-3)+6log_{3}(y) is log_{3}((x-3)^{1/2}y^6)