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

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

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

simplify

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

lo g_{2} (\frac{3}{y}) + \frac{lo g _{2} ( x )}{2}

Solution steps

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

lo g_{2} (9) = 2 lo g_{2} (3)

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

Group like terms

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

Add similar elements:

2 lo g_{2} (3) - lo g_{2} (3) = lo g_{2} (3)

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

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

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

Simplify

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

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

Simplify

\frac{1}{2} lo g_{2} (x) : \frac{lo g _{2} ( x )}{2}

= lo g_{2} (\frac{3}{y}) + \frac{lo g _{2} ( x )}{2}

3 ln (x) - \frac{3}{2} ln (y) + 4 ln (z)

ln (\frac{2 ^{2}}{3 ^{3} 5 ^{4}})

ln (\frac{x x - 9}{x ^{5} + 9})

8 ln (x) - 3 ln (y)

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