# Write log_3 frac{1}{27x^{2} in the form a+b log_3x where a and b are integers

Write ${\mathrm{log}}_{3}\frac{1}{27{x}^{2}}$ in the form a+b ${\mathrm{log}}_{3}x$ where a and b are integers

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krolaniaN

Use the quotient property:
$lo{g}_{b}\left(\frac{m}{n}\right)={\mathrm{log}}_{b}m-{\mathrm{log}}_{b}n$:
$={\mathrm{log}}_{3}1-{\mathrm{log}}_{3}27{x}^{2}$
The logarithm of 1 (using any base) is 0:
$=0-{\mathrm{log}}_{3}27{x}^{2}$
$=-{\mathrm{log}}_{3}27{x}^{2}$
Use the product property: ${\mathrm{log}}_{b}mn={\mathrm{log}}_{b}m+{\mathrm{log}}_{b}n$:
$=-\left({\mathrm{log}}_{3}27+{\mathrm{log}}_{3}{x}^{2}\right)$
$=-\left({\mathrm{log}}_{3}{3}^{3}+lo{g}_{3}{x}^{2}\right)$
Use the rule: ${\mathrm{log}}_{b}{b}^{x}=x$:
$=-\left(3+{\mathrm{log}}_{3}{x}^{2}\right)$
$=-3-{\mathrm{log}}_{3}{x}^{2}$
Use the power property: ${\mathrm{log}}_{b}{x}^{n}=n{\mathrm{log}}_{b}x$
$=-3-2{\mathrm{log}}_{3}x$