What is (1/3 ln|b^3+1|-ln|3b+1|)/3 ?

The solution to (1/3 ln|b^3+1|-ln|3b+1|)/3 is (ln((|b^3+1|^{1/3})/(|3b+1|)))/3

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simplify (1/3 ln|b^3+1|-ln|3b+1|)/3

simplify

\frac{\frac{1}{3} ln b ^{3} + 1 - ln ∣ 3 b + 1 ∣}{3}

\frac{ln ( \frac{∣ b ^{3} + 1 ∣ ^{\frac{1}{3}}}{∣ 3 b + 1 ∣} )}{3}

Solution steps

\frac{\frac{1}{3} ln b ^{3} + 1 - ln ∣ 3 b + 1 ∣}{3}

Apply log rule:

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

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

= \frac{ln ( b ^{3} + 1 ^{\frac{1}{3}} ) - ln ∣ 3 b + 1 ∣}{3}

Apply log rule:

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

ln (b^{3} + 1^{\frac{1}{3}}) - ln ∣ 3 b + 1 ∣ = ln \frac{b ^{3} + 1 ^{\frac{1}{3}}}{∣ 3 b + 1 ∣}

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

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What is (1/3 ln|b^3+1|-ln|3b+1|)/3 ?
The solution to (1/3 ln|b^3+1|-ln|3b+1|)/3 is (ln((|b^3+1|^{1/3})/(|3b+1|)))/3