What is 1/4 log_{4}(y)-1/2 log_{4}(x) ?

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

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simplify 1/4 log_{4}(y)-1/2 log_{4}(x)

simplify

\frac{1}{4} lo g_{4} (y) - \frac{1}{2} lo g_{4} (x)

lo g_{4} (\frac{y ^{\frac{1}{4}} x}{x})

Solution steps

\frac{1}{4} lo g_{4} (y) - \frac{1}{2} lo g_{4} (x)

Apply log rule:

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

\frac{1}{4} lo g_{4} (y) = lo g_{4} (y^{\frac{1}{4}})

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

Apply log rule:

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

\frac{1}{2} lo g_{4} (x) = lo g_{4} (x^{\frac{1}{2}})

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

Apply log rule:

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

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

= lo g_{4} (\frac{y ^{\frac{1}{4}}}{x ^{\frac{1}{2}}})

Simplify

\frac{y ^{\frac{1}{4}}}{x ^{\frac{1}{2}}} : \frac{y ^{\frac{1}{4}} x}{x}

= lo g_{4} (\frac{y ^{\frac{1}{4}} x}{x})

\frac{ln ( 0.24 ) - ln ( 0.12 )}{0.12} \cdot 12

ln (\frac{5}{105})

lo g_{e} (x) + 1 - lo g_{e} (5)

ln (\frac{1}{3 ^{2}})

25 ln (25) - 16 ln (16) - 9 ln (e)

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