What is 1/2 ln|1+x^2|+1/2 ln|1+y^2| ?

The solution to 1/2 ln|1+x^2|+1/2 ln|1+y^2| is ln(|1+x^2|^{1/2}|1+y^2|^{1/2})

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simplify 1/2 ln|1+x^2|+1/2 ln|1+y^2|

simplify

\frac{1}{2} ln 1 + x^{2} + \frac{1}{2} ln 1 + y^{2}

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

Solution steps

\frac{1}{2} ln 1 + x^{2} + \frac{1}{2} ln 1 + y^{2}

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} + 1^{\frac{1}{2}}) + \frac{1}{2} ln y^{2} + 1

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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