What is 1/3 (ln(a)+3ln(b))-1/2 (4ln(c)-2ln(d)) ?

The solution to 1/3 (ln(a)+3ln(b))-1/2 (4ln(c)-2ln(d)) is ln((bd\sqrt[3]{a})/(c^2))

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simplify 1/3 (ln(a)+3ln(b))-1/2 (4ln(c)-2ln(d))

simplify

\frac{1}{3} (ln (a) + 3 ln (b)) - \frac{1}{2} (4 ln (c) - 2 ln (d))

Solution steps

\frac{1}{3} (ln (a) + 3 ln (b)) - \frac{1}{2} (4 ln (c) - 2 ln (d))

Apply log rule:

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

3 ln (b) = ln (b^{3})

= \frac{1}{3} (ln (a) + ln (b^{3})) - \frac{1}{2} (4 ln (c) - 2 ln (d))

Apply log rule:

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

ln (a) + ln (b^{3}) = ln (a b^{3})

= \frac{1}{3} ln (a b^{3}) - \frac{1}{2} (4 ln (c) - 2 ln (d))

Apply log rule:

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

4 ln (c) = ln (c^{4})

= \frac{1}{3} ln (a b^{3}) - \frac{1}{2} (ln (c^{4}) - 2 ln (d))

Apply log rule:

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

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

= \frac{1}{3} ln (a b^{3}) - \frac{1}{2} (ln (c^{4}) - ln (d^{2}))

Apply log rule:

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

ln (c^{4}) - ln (d^{2}) = ln (\frac{c ^{4}}{d ^{2}})

= \frac{1}{3} ln (a b^{3}) - \frac{1}{2} ln (\frac{c ^{4}}{d ^{2}})

Apply log rule:

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

\frac{1}{3} ln (a b^{3}) = ln ((a b^{3})^{\frac{1}{3}})

= ln ((a b^{3})^{\frac{1}{3}}) - \frac{1}{2} ln (\frac{c ^{4}}{d ^{2}})

Apply log rule:

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

\frac{1}{2} ln (\frac{c ^{4}}{d ^{2}}) = ln ((\frac{c ^{4}}{d ^{2}})^{\frac{1}{2}})

= ln ((a b^{3})^{\frac{1}{3}}) - ln ((\frac{c ^{4}}{d ^{2}})^{\frac{1}{2}})

Apply log rule:

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

ln ((a b^{3})^{\frac{1}{3}}) - ln ((\frac{c ^{4}}{d ^{2}})^{\frac{1}{2}}) = ln \frac{( a b ^{3} ) ^{\frac{1}{3}}}{( \frac{c ^{4}}{d ^{2}} ) ^{\frac{1}{2}}}

= ln \frac{( a b ^{3} ) ^{\frac{1}{3}}}{( \frac{c ^{4}}{d ^{2}} ) ^{\frac{1}{2}}}

Simplify

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What is 1/3 (ln(a)+3ln(b))-1/2 (4ln(c)-2ln(d)) ?
The solution to 1/3 (ln(a)+3ln(b))-1/2 (4ln(c)-2ln(d)) is ln((bd\sqrt[3]{a})/(c^2))