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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (a b^{t}) - ln ((ab)^{t}) - 2 ln (a)

ln (\frac{7}{6})

lo g_{10} (0.01) - lo g_{10} (0.001)

lo g_{10} (\frac{y}{4 y + 8})

lo g_{10} ((\frac{y}{3})^{2})

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