Expand Log Expressions

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Question 1 of 3

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$\log_{a}{\frac{xy}{z}}$

1. $log_a x + log_a y - log_a z$
2. $log_a x - log_a y + log_a z$
3. $log_a x + log_a y - 1$
4. $log_x a + log_x y - log_x z$

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Laws of Logarithms

$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$

$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$

Expand the fraction by transforming it into a difference

	$\log_{\color{#9a00c7}{a}} \frac{\color{#00880A}{xy}}{\color{#e65021}{z}}$

$=$	$\log_{\color{#9a00c7}{a}} \color{#00880A}{xy}- \log_{\color{#9a00c7}{a}} \color{#e65021}{z}$

Expand the first term by transforming it into a sum

	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{y}- \log_a z$
$=$	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{y}- \log_a z$

$\log_a x + \log_a y- \log_a z$

Question 2 of 3

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$\log_a x\sqrt{x+4}$

1. $log_x a+log_a (x+4)^(1/2)$
2. $log_x a+log_(x+4) (1/2)$
3. $log_a x+1/2 log_a (x+4)$
4. $log_a (x+4)+1/2 log_a x$

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Laws of Logarithms

$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

Expand the expression by transforming it into a sum

	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{\sqrt{x+4}}$
$=$	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{\sqrt{x+4}}$

Expand further by using the following laws

$\log_b x^\color{#004ec4}{p}$

$=$

$\color{#004ec4}{p}\log_b x$

	$log_a x+log_a sqrt(x+4)$
$=$	$\log_a x+\log_a (x+4)^\color{#CC0000}{\frac{1}{2}}$	Change the surd into an exponent

$=$	$log_a x+$ $1/2$ $log_a (x+4)$	$log_b x^p=p log_b x$

$\log_a x + \frac{1}{2}\log_a (x+4)$

Question 3 of 3

Expand

$\log_{a}{\frac{x(y+z)}{a^3}}$

1. $log_a x + log_a (y+z) - 3$
2. $log_a x + log_a (y+z) + 3$
3. $log_a x + log_a (y+z)^3$
4. $3log_a x +3 log_a (y+z)$

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Laws of Logarithms

$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$

$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} – \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$

Expand the fraction by transforming it into a difference

	$\log_{\color{#9a00c7}{a}} \frac{\color{#00880A}{x(y+z)}}{\color{#e65021}{a^3}}$

$=$	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x(y+z)}-\log_{\color{#9a00c7}{a}} \color{#e65021}{a^3}$

Expand the first term by transforming it into a sum

	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{(y+z)}- \log_a a^3$
$=$	$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{(y+z)}- \log_a a^3$

Expand further by using the following laws

$\log_b x^\color{#004ec4}{p}$	$=$	$\color{#004ec4}{p}\log_b x$
$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}$	$=$	$1$

	$log_a x+log_a (y+z)-log_a a^3$
$=$	$log_a x+log_a (y+z)-$ $3$ $log_a a$	$log_b x^p=p log_b x$
$=$	$log_a x+log_a (y+z)-3($ $1$ $)$	$log_b b=1$
$=$	$log_a x+log_a (y+z)-3$

$log_a x+log_a (y+z)-3$

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Laws of Logarithms

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Laws of Logarithms

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