What is 5log_{5}(x)-1/4 log_{5}(8-x) ?

The solution to 5log_{5}(x)-1/4 log_{5}(8-x) is log_{5}((x^5(8-x)^{3/4})/(8-x))

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simplify 5log_{5}(x)-1/4 log_{5}(8-x)

simplify

5 lo g_{5} (x) - \frac{1}{4} lo g_{5} (8 - x)

lo g_{5} (\frac{x ^{5} ( 8 - x ) ^{\frac{3}{4}}}{8 - x})

Solution steps

5 lo g_{5} (x) - \frac{1}{4} lo g_{5} (8 - x)

Apply log rule:

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

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

= lo g_{5} (x^{5}) - \frac{1}{4} lo g_{5} (- x + 8)

Apply log rule:

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

\frac{1}{4} lo g_{5} (- x + 8) = lo g_{5} ((- x + 8)^{\frac{1}{4}})

= lo g_{5} (x^{5}) - lo g_{5} ((- x + 8)^{\frac{1}{4}})

Apply log rule:

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

lo g_{5} (x^{5}) - lo g_{5} ((8 - x)^{\frac{1}{4}}) = lo g_{5} (\frac{x ^{5}}{( - x + 8 ) ^{\frac{1}{4}}})

= lo g_{5} (\frac{x ^{5}}{( 8 - x ) ^{\frac{1}{4}}})

Simplify

\frac{x ^{5}}{( 8 - x ) ^{\frac{1}{4}}} : \frac{x ^{5} ( 8 - x ) ^{\frac{3}{4}}}{8 - x}

= lo g_{5} (\frac{x ^{5} ( 8 - x ) ^{\frac{3}{4}}}{8 - x})

lo g_{10} (d) - 4 lo g_{10} (g)

lo g_{6} (\frac{2 y - 3}{y})

lo g_{b} (5) + 3 lo g_{b} (10) - lo g_{b} (40)

lo g_{2} (x) + lo g_{2} (x + 2)

lo g_{4} (13 b^{8} c)

What is 5log_{5}(x)-1/4 log_{5}(8-x) ?
The solution to 5log_{5}(x)-1/4 log_{5}(8-x) is log_{5}((x^5(8-x)^{3/4})/(8-x))