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

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

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

simplify

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

lo g_{a} (\frac{x}{( x - y ) ^{3}})

Solution steps

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

Apply log rule:

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

3 lo g_{a} (x - y) = lo g_{a} ((x - y)^{3})

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

Apply log rule:

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

lo g_{a} (x) - lo g_{a} ((x - y)^{3}) = lo g_{a} (\frac{x}{( x - y ) ^{3}})

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

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

Apply log rule:

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

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

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

Simplify

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

= lo g_{a} (\frac{x}{( x - y ) ^{3}})

lo g_{6} (\frac{x ^{7} ( x + 9 )}{( x - 4 ) ^{5}})

ln (x^{2} (x + 6)^{7} (x^{2} + 5)^{4})

1004 \cdot ln (\frac{100}{930}) - 287 \cdot ln (\frac{3}{4})

20 lo g_{10} (\frac{2 \cdot 1 0 ^{- 5}}{1 0 ^{- 6}})

- \frac{1}{9} ln (\frac{e ^{- 18} - 1}{- 18})

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