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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

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

2 ln (x - 1) = ln ((x - 1)^{2})

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

Apply log rule:

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

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

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

Simplify

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

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

lo g_{10} (\frac{y ^{2}}{2 z})

lo g_{10} (18) - lo g_{10} (6)

lo g_{3} (2) + lo g_{3} (13)

2 lo g_{5} (x) - lo g_{5} (y) - \frac{1}{2} lo g_{5} (z)

lo g_{10} (\frac{1000 c ^{5}}{2 d ^{4}})

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