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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

2 lo g_{2} (y) = lo g_{2} (y^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{3} y ^{2}}{z ^{\frac{1}{2}}} : \frac{x ^{3} y ^{2} z}{z}

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

lo g_{7} (\frac{x ^{2} y ^{3}}{z})

ln (\frac{x y}{y ^{3}})

3 ln (a) + 6 ln (b) - ln (8)

lo g_{10} (0^{'} 1)

lo g_{x} (\frac{a ^{3} b}{c})

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