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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (\frac{10}{620.9 9 ^{2}})

\frac{9}{2} ln (2)

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

3 + ln (x) - ln (x + 1)

lo g_{10} (\frac{3}{( x - y ) ^{3}})

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