What is 1/2 ln(z/(2x))+1/2 ln(z/(2y)) ?

The solution to 1/2 ln(z/(2x))+1/2 ln(z/(2y)) is ln((zsqrt(xy))/(2xy))

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simplify 1/2 ln(z/(2x))+1/2 ln(z/(2y))

simplify

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

ln (\frac{z x y}{2 x y})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

(\frac{z}{2 x})^{\frac{1}{2}} (\frac{z}{2 y})^{\frac{1}{2}} : \frac{z x y}{2 x y}

= ln (\frac{z x y}{2 x y})

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What is 1/2 ln(z/(2x))+1/2 ln(z/(2y)) ?
The solution to 1/2 ln(z/(2x))+1/2 ln(z/(2y)) is ln((zsqrt(xy))/(2xy))