What is 3log_{5}(x)-1/2 log_{5}(6-x) ?

The solution to 3log_{5}(x)-1/2 log_{5}(6-x) is log_{5}((x^3sqrt(6-x))/(6-x))

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simplify 3log_{5}(x)-1/2 log_{5}(6-x)

simplify

3 lo g_{5} (x) - \frac{1}{2} lo g_{5} (6 - x)

lo g_{5} (\frac{x ^{3} 6 - x}{6 - x})

Solution steps

3 lo g_{5} (x) - \frac{1}{2} lo g_{5} (6 - x)

Apply log rule:

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

3 lo g_{5} (x) = lo g_{5} (x^{3})

= lo g_{5} (x^{3}) - \frac{1}{2} lo g_{5} (- x + 6)

Apply log rule:

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

\frac{1}{2} lo g_{5} (- x + 6) = lo g_{5} ((- x + 6)^{\frac{1}{2}})

= lo g_{5} (x^{3}) - lo g_{5} ((- x + 6)^{\frac{1}{2}})

Apply log rule:

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

lo g_{5} (x^{3}) - lo g_{5} ((6 - x)^{\frac{1}{2}}) = lo g_{5} (\frac{x ^{3}}{( - x + 6 ) ^{\frac{1}{2}}})

= lo g_{5} (\frac{x ^{3}}{( 6 - x ) ^{\frac{1}{2}}})

Simplify

\frac{x ^{3}}{( 6 - x ) ^{\frac{1}{2}}} : \frac{x ^{3} 6 - x}{6 - x}

= lo g_{5} (\frac{x ^{3} 6 - x}{6 - x})

ln (x) + ln (20)

lo g_{5} (5 x^{2})

lo g_{2} (5) + lo g_{2} (7)

ln (\frac{27}{64})

6 ln (x) - \frac{1}{3} ln (y)

What is 3log_{5}(x)-1/2 log_{5}(6-x) ?
The solution to 3log_{5}(x)-1/2 log_{5}(6-x) is log_{5}((x^3sqrt(6-x))/(6-x))