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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

ln (\frac{11}{3})

ln (\frac{e}{a})

ln (\frac{p}{1 - p})

ln (\frac{60}{70})

ln (\frac{1}{1 - 0.3})

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