What is 4ln(x)-1/9 ln(y) ?

The solution to 4ln(x)-1/9 ln(y) is ln((x^4y^{8/9})/y)

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simplify 4ln(x)-1/9 ln(y)

simplify

4 ln (x) - \frac{1}{9} ln (y)

ln (\frac{x ^{4} y ^{\frac{8}{9}}}{y})

Solution steps

4 ln (x) - \frac{1}{9} ln (y)

Apply log rule:

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

4 ln (x) = ln (x^{4})

= ln (x^{4}) - \frac{1}{9} ln (y)

Apply log rule:

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

\frac{1}{9} ln (y) = ln (y^{\frac{1}{9}})

= ln (x^{4}) - ln (y^{\frac{1}{9}})

Apply log rule:

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

ln (x^{4}) - ln (y^{\frac{1}{9}}) = ln (\frac{x ^{4}}{y ^{\frac{1}{9}}})

= ln (\frac{x ^{4}}{y ^{\frac{1}{9}}})

Simplify

\frac{x ^{4}}{y ^{\frac{1}{9}}} : \frac{x ^{4} y ^{\frac{8}{9}}}{y}

= ln (\frac{x ^{4} y ^{\frac{8}{9}}}{y})

lo g_{3} (2 x)

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

lo g_{7} (343 x)

\frac{1}{2} (lo g_{6} (s) + lo g_{6} (t))

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

What is 4ln(x)-1/9 ln(y) ?
The solution to 4ln(x)-1/9 ln(y) is ln((x^4y^{8/9})/y)