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

Question

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

Suman Cole 2021-10-20 Answered

Condense the following log expression.
$- 2 \log_{3} (x y) + \frac{4}{5} \log_{3} z^{10} + \log_{3} (x + 1)$

$x \log_{a} b = \log_{a} b^{x}$

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Expert Answer

averes8

Answered 2021-10-21 Author has 92 answers

Step 1
Condensing the logarithmic expression.
Given:
$- 2 \log_{3} (x y) + \frac{4}{5} \log_{3} z^{10} + \log_{3} (x + 1)$

$x \log_{a} b = \log_{a} b^{x}$
Apply log rule: $x \log_{a} b = \log_{a} b^{x}$
$- \log_{3} {(x y)}^{2} + \log_{3} {(z^{10})}^{\frac{4}{5}} + \log_{3} (x + 1)$
Step 2
By the property of logarithms.
$\log_{a} (c d) = \log_{a} c + \log_{a} d$
$\Rightarrow - \log_{3} {(x y)}^{2} + \log_{3} (z^{8}) + \log_{3} (x + 1)$
$\Rightarrow - \log_{3} {(x y)}^{2} + \log_{3} (z^{8} (x + 1))$
Step 3
By the property of logarithms.
$\log_{a} (\frac{c}{d}) = \log_{a} c - \log_{a} d$
$\Rightarrow \log_{3} (z^{8} (x + 1)) - \log_{3} {(x y)}^{2}$
$\Rightarrow \log_{3} (\frac{z^{8} (x + 1)}{{(x y)}^{2}})$