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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

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

ln (2) + ln (9)

3 ln (x + 4) + 9 ln (x - 4) - 5 ln (x)

lo g_{4} (\frac{y}{5 z ^{2}})

- ln (y) - ln (x)

\frac{1}{3} ln (x + 8) + \frac{1}{2} ln (x - 1)

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