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

The solution to 1/3 ln(16p)-3ln(2p)-2ln(5)p^4+6ln(q) is ln((q^6)/(2*2^{2/3)p^{8/3}})-2ln(5)p^4

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

simplify

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

ln (\frac{q ^{6}}{2 \cdot 2 ^{\frac{2}{3}} p ^{\frac{8}{3}}}) - 2 ln (5) p^{4}

Solution steps

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

Apply log rule:

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

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

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

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

Simplify

Apply log rule:

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

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

Apply log rule:

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

Simplify

= ln (\frac{q ^{6}}{2 ^{1 + \frac{2}{3}} p ^{\frac{8}{3}}}) - 2 ln (5) p^{4}

\frac{q ^{6}}{2 ^{1 + \frac{2}{3}} p ^{\frac{8}{3}}} = \frac{q ^{6}}{2 \cdot 2 ^{\frac{2}{3}} p ^{\frac{8}{3}}}

= ln (\frac{q ^{6}}{2 \cdot 2 ^{\frac{2}{3}} p ^{\frac{8}{3}}}) - 2 ln (5) p^{4}

ln (\frac{1}{\frac{2}{2}})

lo g_{7} ((x y^{5})^{2})

ln (\frac{800 - 1200}{300 - 1200})

0.77 + \frac{8.134 \cdot 298}{1 \cdot 96500} \cdot ln (\frac{1}{2})

(ln (2) + ln (3)) \frac{π}{8}

What is 1/3 ln(16p)-3ln(2p)-2ln(5)p^4+6ln(q) ?
The solution to 1/3 ln(16p)-3ln(2p)-2ln(5)p^4+6ln(q) is ln((q^6)/(2*2^{2/3)p^{8/3}})-2ln(5)p^4