What is 3log_{9}(x)-1/2 log_{9}(y)-5log_{9}(z-4) ?

The solution to 3log_{9}(x)-1/2 log_{9}(y)-5log_{9}(z-4) is log_{9}((x^3sqrt(y))/(y(z-4)^5))

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simplify 3log_{9}(x)-1/2 log_{9}(y)-5log_{9}(z-4)

simplify

3 lo g_{9} (x) - \frac{1}{2} lo g_{9} (y) - 5 lo g_{9} (z - 4)

lo g_{9} (\frac{x ^{3} y}{y ( z - 4 ) ^{5}})

Solution steps

3 lo g_{9} (x) - \frac{1}{2} lo g_{9} (y) - 5 lo g_{9} (z - 4)

Apply log rule:

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

3 lo g_{9} (x) = lo g_{9} (x^{3})

= lo g_{9} (x^{3}) - \frac{1}{2} lo g_{9} (y) - 5 lo g_{9} (z - 4)

Apply log rule:

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

\frac{1}{2} lo g_{9} (y) = lo g_{9} (y^{\frac{1}{2}})

= lo g_{9} (x^{3}) - lo g_{9} (y^{\frac{1}{2}}) - 5 lo g_{9} (z - 4)

Apply log rule:

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

lo g_{9} (x^{3}) - lo g_{9} (y^{\frac{1}{2}}) = lo g_{9} (\frac{x ^{3}}{y ^{\frac{1}{2}}})

= lo g_{9} (\frac{x ^{3}}{y ^{\frac{1}{2}}}) - 5 lo g_{9} (z - 4)

Simplify

\frac{x ^{3}}{y ^{\frac{1}{2}}} : \frac{x ^{3} y}{y}

= lo g_{9} (\frac{x ^{3} y}{y}) - 5 lo g_{9} (z - 4)

Apply log rule:

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

5 lo g_{9} (z - 4) = lo g_{9} ((z - 4)^{5})

= lo g_{9} (\frac{x ^{3} y}{y}) - lo g_{9} ((z - 4)^{5})

Apply log rule:

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

lo g_{9} (\frac{x ^{3} y}{y}) - lo g_{9} ((z - 4)^{5}) = lo g_{9} \frac{\frac{x ^{3} y}{y}}{( z - 4 ) ^{5}}

= lo g_{9} \frac{\frac{x ^{3} y}{y}}{( z - 4 ) ^{5}}

Apply the fraction rule:

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

= lo g_{9} (\frac{x ^{3} y}{y ( z - 4 ) ^{5}})

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What is 3log_{9}(x)-1/2 log_{9}(y)-5log_{9}(z-4) ?
The solution to 3log_{9}(x)-1/2 log_{9}(y)-5log_{9}(z-4) is log_{9}((x^3sqrt(y))/(y(z-4)^5))