What is ln(x)-1/2 ln(4+v^2) ?

The solution to ln(x)-1/2 ln(4+v^2) is ln((xsqrt(4+v^2))/(4+v^2))

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simplify ln(x)-1/2 ln(4+v^2)

simplify

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

ln (\frac{x 4 + v ^{2}}{4 + v ^{2}})

Solution steps

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

Apply log rule:

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

\frac{1}{2} ln (v^{2} + 4) = ln ((v^{2} + 4)^{\frac{1}{2}})

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

Apply log rule:

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

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

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

Simplify

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

= ln (\frac{x 4 + v ^{2}}{4 + v ^{2}})

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What is ln(x)-1/2 ln(4+v^2) ?
The solution to ln(x)-1/2 ln(4+v^2) is ln((xsqrt(4+v^2))/(4+v^2))