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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

4 lo g_{10} (\frac{x}{5})

- lo g_{10} (6.5 \cdot 1 0^{- 5})

lo g_{3} (x) + lo g_{3} (x + 8)

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