# Condense the following log expression.-2\log_3(xy)+\frac{4}{5}\log_3z^{1

Condense the following log expression.
$-2{\mathrm{log}}_{3}\left(xy\right)+\frac{4}{5}{\mathrm{log}}_{3}{z}^{10}+{\mathrm{log}}_{3}\left(x+1\right)$

$x{\mathrm{log}}_{a}b={\mathrm{log}}_{a}{b}^{x}$

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Step 1
Condensing the logarithmic expression.
Given:
$-2{\mathrm{log}}_{3}\left(xy\right)+\frac{4}{5}{\mathrm{log}}_{3}{z}^{10}+{\mathrm{log}}_{3}\left(x+1\right)$

$x{\mathrm{log}}_{a}b={\mathrm{log}}_{a}{b}^{x}$
Apply log rule: $x{\mathrm{log}}_{a}b={\mathrm{log}}_{a}{b}^{x}$
$-{\mathrm{log}}_{3}{\left(xy\right)}^{2}+{\mathrm{log}}_{3}{\left({z}^{10}\right)}^{\frac{4}{5}}+{\mathrm{log}}_{3}\left(x+1\right)$
Step 2
By the property of logarithms.
${\mathrm{log}}_{a}\left(cd\right)={\mathrm{log}}_{a}c+{\mathrm{log}}_{a}d$
$⇒-{\mathrm{log}}_{3}{\left(xy\right)}^{2}+{\mathrm{log}}_{3}\left({z}^{8}\right)+{\mathrm{log}}_{3}\left(x+1\right)$
$⇒-{\mathrm{log}}_{3}{\left(xy\right)}^{2}+{\mathrm{log}}_{3}\left({z}^{8}\left(x+1\right)\right)$
Step 3
By the property of logarithms.
${\mathrm{log}}_{a}\left(\frac{c}{d}\right)={\mathrm{log}}_{a}c-{\mathrm{log}}_{a}d$
$⇒{\mathrm{log}}_{3}\left({z}^{8}\left(x+1\right)\right)-{\mathrm{log}}_{3}{\left(xy\right)}^{2}$
$⇒{\mathrm{log}}_{3}\left(\frac{{z}^{8}\left(x+1\right)}{{\left(xy\right)}^{2}}\right)$