What is 1/2 log_{3}(x)+1/3 log_{3}(2x+9) ?

The solution to 1/2 log_{3}(x)+1/3 log_{3}(2x+9) is log_{3}(x^{1/2}(2x+9)^{1/3})

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simplify 1/2 log_{3}(x)+1/3 log_{3}(2x+9)

simplify

\frac{1}{2} lo g_{3} (x) + \frac{1}{3} lo g_{3} (2 x + 9)

lo g_{3} (x^{\frac{1}{2}} (2 x + 9)^{\frac{1}{3}})

Solution steps

\frac{1}{2} lo g_{3} (x) + \frac{1}{3} lo g_{3} (2 x + 9)

Apply log rule:

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

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

= lo g_{3} (x^{\frac{1}{2}}) + \frac{1}{3} lo g_{3} (2 x + 9)

Apply log rule:

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

\frac{1}{3} lo g_{3} (2 x + 9) = lo g_{3} ((2 x + 9)^{\frac{1}{3}})

= lo g_{3} (x^{\frac{1}{2}}) + lo g_{3} ((2 x + 9)^{\frac{1}{3}})

Apply log rule:

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

lo g_{3} (x^{\frac{1}{2}}) + lo g_{3} ((2 x + 9)^{\frac{1}{3}}) = lo g_{3} (x^{\frac{1}{2}} (2 x + 9)^{\frac{1}{3}})

= lo g_{3} (x^{\frac{1}{2}} (2 x + 9)^{\frac{1}{3}})

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What is 1/2 log_{3}(x)+1/3 log_{3}(2x+9) ?
The solution to 1/2 log_{3}(x)+1/3 log_{3}(2x+9) is log_{3}(x^{1/2}(2x+9)^{1/3})