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

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

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

simplify

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

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

Solution steps

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

ln (\frac{6 x ^{2}}{y ^{4}})

lo g_{10} (lo g_{10} (1000 0^{25 x}))

ln (3 x + 1) - ln (x + 2)

lo g_{3} (\frac{ab}{c d})

lo g_{10} (rs)

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