What is 1/3 ln(16p)-3ln(2pq)-2ln(5p^4)+6ln(q) ?

The solution to 1/3 ln(16p)-3ln(2pq)-2ln(5p^4)+6ln(q) is ln((\sqrt[3]{2}q^3)/(2^2*5^2p^{32/3)})

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simplify 1/3 ln(16p)-3ln(2pq)-2ln(5p^4)+6ln(q)

simplify

\frac{1}{3} ln (16 p) - 3 ln (2 pq) - 2 ln (5 p^{4}) + 6 ln (q)

ln (\frac{3 2 q ^{3}}{2 ^{2} \cdot 5 ^{2} p ^{\frac{32}{3}}})

Solution steps

\frac{1}{3} ln (16 p) - 3 ln (2 pq) - 2 ln (5 p^{4}) + 6 ln (q)

Apply log rule:

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

\frac{1}{3} ln (16 p) = ln ((16 p)^{\frac{1}{3}})

= ln ((16 p)^{\frac{1}{3}}) - 3 ln (2 pq) - 2 ln (5 p^{4}) + 6 ln (q)

Apply log rule:

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

3 ln (2 pq) = ln ((2 pq)^{3})

= ln ((16 p)^{\frac{1}{3}}) - ln ((2 pq)^{3}) - 2 ln (5 p^{4}) + 6 ln (q)

Apply log rule:

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

ln ((16 p)^{\frac{1}{3}}) - ln ((2 pq)^{3}) = ln (\frac{( 16 p ) ^{\frac{1}{3}}}{( 2 pq ) ^{3}})

= ln (\frac{( 16 p ) ^{\frac{1}{3}}}{( 2 pq ) ^{3}}) - 2 ln (5 p^{4}) + 6 ln (q)

Simplify

\frac{( 16 p ) ^{\frac{1}{3}}}{( 2 pq ) ^{3}} : \frac{3 2 3 p}{4 p ^{3} q ^{3}}

= ln (\frac{3 2 3 p}{4 p ^{3} q ^{3}}) - 2 ln (5 p^{4}) + 6 ln (q)

Apply log rule:

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

2 ln (5 p^{4}) = ln ((5 p^{4})^{2})

= ln (\frac{3 2 3 p}{4 p ^{3} q ^{3}}) - ln ((5 p^{4})^{2}) + 6 ln (q)

Apply log rule:

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

ln (\frac{3 2 3 p}{4 p ^{3} q ^{3}}) - ln ((5 p^{4})^{2}) = ln \frac{\frac{3 2 3 p}{4 p ^{3} q ^{3}}}{( 5 p ^{4} ) ^{2}}

= ln \frac{\frac{3 2 3 p}{4 p ^{3} q ^{3}}}{( 5 p ^{4} ) ^{2}} + 6 ln (q)

Simplify

\frac{\frac{3 2 3 p}{4 p ^{3} q ^{3}}}{( 5 p ^{4} ) ^{2}} : \frac{3 2 3 p}{100 p ^{11} q ^{3}}

= ln (\frac{3 2 3 p}{100 p ^{11} q ^{3}}) + 6 ln (q)

Apply log rule:

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

6 ln (q) = ln (q^{6})

= ln (\frac{3 2 3 p}{100 p ^{11} q ^{3}}) + ln (q^{6})

Apply log rule:

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

ln (\frac{3 2 3 p}{100 p ^{11} q ^{3}}) + ln (q^{6}) = ln (q^{6} \frac{3 2 3 p}{100 p ^{11} q ^{3}})

= ln (q^{6} \frac{3 2 3 p}{100 p ^{11} q ^{3}})

Simplify

\frac{3 2 3 p}{100 p ^{11} q ^{3}} q^{6} : \frac{3 2 q ^{3}}{100 p ^{\frac{32}{3}}}

= ln (\frac{3 2 q ^{3}}{100 p ^{\frac{32}{3}}})

\frac{3 2 q ^{3}}{100 p ^{\frac{32}{3}}} = \frac{3 2 q ^{3}}{2 ^{2} \cdot 5 ^{2} p ^{\frac{32}{3}}}

= ln (\frac{3 2 q ^{3}}{2 ^{2} \cdot 5 ^{2} p ^{\frac{32}{3}}})

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What is 1/3 ln(16p)-3ln(2pq)-2ln(5p^4)+6ln(q) ?
The solution to 1/3 ln(16p)-3ln(2pq)-2ln(5p^4)+6ln(q) is ln((\sqrt[3]{2}q^3)/(2^2*5^2p^{32/3)})