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Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm.

Can anyone help me with this question? Thanks

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2 Answers

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Using properties of logarithms we have

$$a\ln b=\ln b^a$$ and $$\ln x+\ln y=\ln xy$$

We get

$$\begin{align} 2 \ln (2 - x) + 3 \ln (x^2 - 5)&= \ln (2 - x)^2 + \ln (x^2 - 5)^3\\&=\ln \Big[(2 - x)^2(x^2 -5)^3\Big]\\\end{align}$$

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$$\large\color{blue}2 \ln \color{green}{(2 - x)} + \color{red}3 \ln \color{orange}{(x^2 - 5)} =\ln\color{green}{(2 - x)}^{\color{blue}2}\color{orange}{(x^2 - 5)}^{\color{red}3}$$

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