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

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

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

simplify

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

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

Solution steps

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

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

Simplify

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

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

ln (3 x)

4 ln (3) - 4 ln (2)

lo g_{10} (\frac{( 2 x ^{2} + 6 )}{( 3 x + 9 )})

ln (\frac{a x + b}{c x + d})

\frac{1}{4} \cdot lo g_{e} (\frac{4}{3})

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