Gage Potter

2022-04-25

Using laws of logarithms, write the expression in the image as a single logarithm.

$6\mathrm{log}\left(64\u2013{x}^{2}\right)\u2013(\mathrm{log}(8+x)+\mathrm{log}\left(8\u2013x\right)=\mathrm{log}()$

Aliana Sexton

Beginner2022-04-26Added 20 answers

Apply law of logarithm

$\mathrm{log}m+\mathrm{log}n=\mathrm{log}mn$

$\mathrm{log}m-\mathrm{log}n=\mathrm{log}\frac{m}{n}$

$6\mathrm{log}\left(64\u2013{x}^{2}\right)\u2013(\mathrm{log}(8+x)+\mathrm{log}\left(8\u2013x\right)=\mathrm{log}()$

$6\mathrm{log}({8}^{2}-{x}^{2})-\mathrm{log}(8+x)-\mathrm{log}(8-x)$

$6\mathrm{log}(8-x)(8+x)-\mathrm{log}(8+x)-\mathrm{log}(8-x)$

$\Rightarrow 6\mathrm{log}(8-x)+6\mathrm{log}(8+x)-\mathrm{log}(8-x)$

$\Rightarrow 5\mathrm{log}(8-x)+5\mathrm{log}(8+x)$

$\Rightarrow 5\mathrm{log}(8-x)(8+x)$

$\Rightarrow 5\mathrm{log}({8}^{2}-{x}^{2})$

$\Rightarrow 5\mathrm{log}(64-{x}^{2})$

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