What is ln(x)-2ln(z)-1/6 ln(y) ?

The solution to ln(x)-2ln(z)-1/6 ln(y) is ln((xy^{5/6})/(z^2y))

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simplify ln(x)-2ln(z)-1/6 ln(y)

simplify

ln (x) - 2 ln (z) - \frac{1}{6} ln (y)

ln (\frac{x y ^{\frac{5}{6}}}{z ^{2} y})

Solution steps

ln (x) - 2 ln (z) - \frac{1}{6} ln (y)

Apply log rule:

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

2 ln (z) = ln (z^{2})

= ln (x) - ln (z^{2}) - \frac{1}{6} ln (y)

Apply log rule:

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

ln (x) - ln (z^{2}) = ln (\frac{x}{z ^{2}})

= ln (\frac{x}{z ^{2}}) - \frac{1}{6} ln (y)

Apply log rule:

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

\frac{1}{6} ln (y) = ln (y^{\frac{1}{6}})

= ln (\frac{x}{z ^{2}}) - ln (y^{\frac{1}{6}})

Apply log rule:

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

ln (\frac{x}{z ^{2}}) - ln (y^{\frac{1}{6}}) = ln (\frac{\frac{x}{z ^{2}}}{y ^{\frac{1}{6}}})

= ln (\frac{\frac{x}{z ^{2}}}{y ^{\frac{1}{6}}})

Simplify

\frac{\frac{x}{z ^{2}}}{y ^{\frac{1}{6}}} : \frac{x y ^{\frac{5}{6}}}{z ^{2} y}

= ln (\frac{x y ^{\frac{5}{6}}}{z ^{2} y})

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What is ln(x)-2ln(z)-1/6 ln(y) ?
The solution to ln(x)-2ln(z)-1/6 ln(y) is ln((xy^{5/6})/(z^2y))