# Express the given expression as a single logarithm Express 2ln(2−x)+3ln(x^2−5)

Express the given expression as a single logarithm
Express
$2\mathrm{ln}\left(2-x\right)+3\mathrm{ln}\left({x}^{2}-5\right)$
as a single logarithm.
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Olleschauvh
Using properties of logarithms we have
$a\mathrm{ln}b=\mathrm{ln}{b}^{a}$
and
$\mathrm{ln}x+\mathrm{ln}y=\mathrm{ln}xy$
We get
$\begin{array}{rl}2\mathrm{ln}\left(2-x\right)+3\mathrm{ln}\left({x}^{2}-5\right)& =\mathrm{ln}\left(2-x{\right)}^{2}+\mathrm{ln}\left({x}^{2}-5{\right)}^{3}\\ & =\mathrm{ln}\left[\left(2-x{\right)}^{2}\left({x}^{2}-5{\right)}^{3}\right]\end{array}$
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Tatiana Cook
$2\mathrm{ln}\left(2-x\right)+3\mathrm{ln}\left({x}^{2}-5\right)=\mathrm{ln}{\left(2-x\right)}^{2}{\left({x}^{2}-5\right)}^{3}$