What is 2ln(3)+ln(x)-1/3 ln(x^4+1) ?

The solution to 2ln(3)+ln(x)-1/3 ln(x^4+1) is ln((9x(x^4+1)^{2/3})/(x^4+1))

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simplify 2ln(3)+ln(x)-1/3 ln(x^4+1)

simplify

2 ln (3) + ln (x) - \frac{1}{3} ln (x^{4} + 1)

ln \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}

Solution steps

2 ln (3) + ln (x) - \frac{1}{3} ln (x^{4} + 1)

Apply log rule:

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

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

= ln (3^{2}) + ln (x) - \frac{1}{3} ln (x^{4} + 1)

Apply log rule:

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

ln (3^{2}) + ln (x) = ln (3^{2} x)

= ln (3^{2} x) - \frac{1}{3} ln (x^{4} + 1)

3^{2} = 9

= ln (9 x) - \frac{1}{3} ln (x^{4} + 1)

Apply log rule:

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

\frac{1}{3} ln (x^{4} + 1) = ln ((x^{4} + 1)^{\frac{1}{3}})

= ln (9 x) - ln ((x^{4} + 1)^{\frac{1}{3}})

Apply log rule:

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

ln (9 x) - ln ((x^{4} + 1)^{\frac{1}{3}}) = ln (\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}})

= ln (\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}})

Simplify

\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}} : \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}

= ln \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}

ln (\frac{8}{15})

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What is 2ln(3)+ln(x)-1/3 ln(x^4+1) ?
The solution to 2ln(3)+ln(x)-1/3 ln(x^4+1) is ln((9x(x^4+1)^{2/3})/(x^4+1))