How do you expand $ln (\sqrt{(\frac{x y^{2}}{z})}$ $ln(sqrt(((xy^2)/z))$ ?

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Answer 1 · 2016-05-17T00:45:01

For this problem, we will use the following $ln$ $ln$ properties:

$∙ {ln}^{a} = a ln$ $• ln^a = aln$
$∙ ln (\frac{x}{y}) = ln x - ln y$ $• ln(x/y) = lnx - lny$
$∙ ln (x y) = ln x + ln y$ $• ln(xy) = lnx + lny$

First, let's establish that $\sqrt{a} = a^{\frac{1}{2}}$ $sqrt(a) = a^(1/2)$

Therefore, $ln (\sqrt{\frac{x y^{2}}{z}}) = {ln (\frac{x y^{2}}{z})}^{\frac{1}{2}}$ $ln(sqrt((xy^2)/z)) = ln((xy^2)/z)^(1/2)$

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

$= \frac{1}{2} (ln x y^{2} - ln z)$ $= 1/2(lnxy^2 - lnz)$

$= \frac{1}{2} (ln x + {ln y}^{2} - ln z)$ $= 1/2(lnx + lny^2 - lnz)$

Hopefully this helps!

How do you expand ln(√(xy2z)ln(sqrt(((xy^2)/z))?