What is 1/2 log_{a}(x)-8log_{a}(y)+log_{a}(z) ?

The solution to 1/2 log_{a}(x)-8log_{a}(y)+log_{a}(z) is log_{a}((zx^{1/2})/(y^8))

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simplify 1/2 log_{a}(x)-8log_{a}(y)+log_{a}(z)

simplify

\frac{1}{2} lo g_{a} (x) - 8 lo g_{a} (y) + lo g_{a} (z)

lo g_{a} (\frac{z x ^{\frac{1}{2}}}{y ^{8}})

Solution steps

\frac{1}{2} lo g_{a} (x) - 8 lo g_{a} (y) + lo g_{a} (z)

Apply log rule:

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

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

= lo g_{a} (x^{\frac{1}{2}}) - 8 lo g_{a} (y) + lo g_{a} (z)

Apply log rule:

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

8 lo g_{a} (y) = lo g_{a} (y^{8})

= lo g_{a} (x^{\frac{1}{2}}) - lo g_{a} (y^{8}) + lo g_{a} (z)

Apply log rule:

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

lo g_{a} (x^{\frac{1}{2}}) - lo g_{a} (y^{8}) = lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{8}})

= lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{8}}) + lo g_{a} (z)

Apply log rule:

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

lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{8}}) + lo g_{a} (z) = lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{8}} z)

= lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{8}} z)

Multiply fractions:

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

= lo g_{a} (\frac{z x ^{\frac{1}{2}}}{y ^{8}})

ln (42)

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

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

lo g_{10} (x y^{4})

lo g_{5} (7) + lo g_{5} (3)

What is 1/2 log_{a}(x)-8log_{a}(y)+log_{a}(z) ?
The solution to 1/2 log_{a}(x)-8log_{a}(y)+log_{a}(z) is log_{a}((zx^{1/2})/(y^8))