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

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

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

simplify

2 lo g_{3} (3 x) - lo g_{3} (2 x) + lo g_{3} (x^{2} + 1)

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

Solution steps

2 lo g_{3} (3 x) - lo g_{3} (2 x) + lo g_{3} (x^{2} + 1)

Simplify

2 lo g_{3} (3 x) : 2 + 2 lo g_{3} (x)

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

Apply log rule:

b \cdot lo g_{a} (x) = lo g_{a} (x^{b})

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

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

Simplify

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

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

lo g_{10} (x^{2} + 3 x + 2) - lo g_{10} (x^{2} - 4)

lo g_{6} (x^{4} y^{3})

lo g_{a} (0.78) - lo g_{a} (6)

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

- ln (\frac{2}{2}) + ln (1)

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