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

The solution to ln(x)-1/2 ln(1+x^2) is ln((xsqrt(1+x^2))/(1+x^2))

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

simplify

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

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

Solution steps

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

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

Simplify

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

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

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