What is 2ln(p/q)+1/2 ln(q^2)-ln(p^3) ?

The solution to 2ln(p/q)+1/2 ln(q^2)-ln(p^3) is -ln(pq)

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simplify 2ln(p/q)+1/2 ln(q^2)-ln(p^3)

simplify

2 ln (\frac{p}{q}) + \frac{1}{2} ln (q^{2}) - ln (p^{3})

- ln (pq)

Solution steps

2 ln (\frac{p}{q}) + \frac{1}{2} ln (q^{2}) - ln (p^{3})

Apply log rule:

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

2 ln (\frac{p}{q}) = ln ((\frac{p}{q})^{2})

= ln ((\frac{p}{q})^{2}) + \frac{1}{2} ln (q^{2}) - ln (p^{3})

Apply log rule:

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

\frac{1}{2} ln (q^{2}) = ln ((q^{2})^{\frac{1}{2}})

= ln ((\frac{p}{q})^{2}) + ln ((q^{2})^{\frac{1}{2}}) - ln (p^{3})

Apply log rule:

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

ln ((\frac{p}{q})^{2}) + ln ((q^{2})^{\frac{1}{2}}) = ln ((\frac{p}{q})^{2} (q^{2})^{\frac{1}{2}})

= ln ((\frac{p}{q})^{2} (q^{2})^{\frac{1}{2}}) - ln (p^{3})

Simplify

(\frac{p}{q})^{2} (q^{2})^{\frac{1}{2}} : \frac{p ^{2}}{q}

= ln (\frac{p ^{2}}{q}) - ln (p^{3})

Apply log rule:

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

ln (\frac{p ^{2}}{q}) - ln (p^{3}) = ln \frac{\frac{p ^{2}}{q}}{p ^{3}}

= ln \frac{\frac{p ^{2}}{q}}{p ^{3}}

Simplify

\frac{\frac{p ^{2}}{q}}{p ^{3}} : \frac{1}{pq}

= ln (\frac{1}{pq})

Apply log rule:

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

= - ln (pq)

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What is 2ln(p/q)+1/2 ln(q^2)-ln(p^3) ?
The solution to 2ln(p/q)+1/2 ln(q^2)-ln(p^3) is -ln(pq)