What is ln(x/(y^3))-3*1/y+1/2*ln(1/(x^4y^2)) ?

The solution to ln(x/(y^3))-3*1/y+1/2*ln(1/(x^4y^2)) is -ln(xy^4)-3/y

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simplify ln(x/(y^3))-31/y+1/2ln(1/(x^4y^2))

simplify

ln (\frac{x}{y ^{3}}) - 3 \cdot \frac{1}{y} + \frac{1}{2} \cdot ln (\frac{1}{x ^{4} y ^{2}})

- ln (x y^{4}) - \frac{3}{y}

Solution steps

ln (\frac{x}{y ^{3}}) - 3 \cdot \frac{1}{y} + \frac{1}{2} ln (\frac{1}{x ^{4} y ^{2}})

Apply log rule:

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

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

= ln (\frac{x}{y ^{3}}) - 3 \cdot \frac{1}{y} + ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (\frac{x}{y ^{3}}) + ln ((\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}}) = ln (\frac{x}{y ^{3}} (\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}})

= ln (\frac{x}{y ^{3}} (\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}}) - 3 \cdot \frac{1}{y}

Simplify

\frac{x}{y ^{3}} (\frac{1}{x ^{4} y ^{2}})^{\frac{1}{2}} : \frac{1}{x y ^{4}}

= ln (\frac{1}{x y ^{4}}) - 3 \cdot \frac{1}{y}

3 \cdot \frac{1}{y} = \frac{3}{y}

= ln (\frac{1}{x y ^{4}}) - \frac{3}{y}

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (x y^{4}) - \frac{3}{y}

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What is ln(x/(y^3))-3*1/y+1/2*ln(1/(x^4y^2)) ?
The solution to ln(x/(y^3))-3*1/y+1/2*ln(1/(x^4y^2)) is -ln(xy^4)-3/y