logosdepmpe

Answered

2022-08-11

use the laws of logarithms to expand the expression
${\mathrm{log}}_{3}\left(\frac{x\left({x}^{2}+5\right)}{\sqrt{{x}^{2}-5}}\right)$

Answer & Explanation

Kyle George

Expert

2022-08-12Added 22 answers

${\mathrm{log}}_{3}\left(\frac{x\left({x}^{2}+5\right)}{\sqrt{{x}^{2}-5}}\right)\phantom{\rule{0ex}{0ex}}⇒{\mathrm{log}}_{3}\left(x\left({x}^{2}+5\right)\right)-{\mathrm{log}}_{3}\sqrt{{x}^{2}-5}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{log}}_{3}x+{\mathrm{log}}_{3}\left({x}^{2}+5\right)-{\mathrm{log}}_{3}\left({x}^{2}-5{\right)}^{1/2}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{log}}_{3}x+{\mathrm{log}}_{3}\left({x}^{2}+5\right)-\frac{1}{2}{\mathrm{log}}_{3}\left({x}^{2}-5\right)$
$\left\{\begin{array}{l}\mathrm{log}\left(m\cdot n\right)=\mathrm{log}m+\mathrm{log}n\\ \mathrm{log}\left(m{\right)}^{n}=n\mathrm{log}m\end{array}$
Ans.
inter chang x and y
$x=\sqrt{3+7y}\phantom{\rule{0ex}{0ex}}{x}^{2}-3=7y\phantom{\rule{0ex}{0ex}}y=\frac{{x}^{2}-3}{7}\phantom{\rule{0ex}{0ex}}{f}^{-1}\left(x\right)=\frac{{x}^{2}-3}{7}$

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