Expanding Log Expressions

Question 1 of 3

Expand

{log}_{a} \frac{x y}{z}

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

1. ${log}_{x} a + {log}_{x} y - {log}_{x} z$ $log_x a + log_x y - log_x z$
2. ${log}_{a} x + {log}_{a} y - 1$ $log_a x + log_a y - 1$
3. ${log}_{a} x - {log}_{a} y + {log}_{a} z$ $log_a x - log_a y + log_a z$
4. ${log}_{a} x + {log}_{a} y - {log}_{a} z$ $log_a x + log_a y - log_a z$

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Laws of Logarithms

{log}_{b} x y = {log}_{b} x + {log}_{b} y

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

{log}_{b} \frac{x}{y} = {log}_{b} x - {log}_{b} 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

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

Question 2 of 3

Expand

{log}_{a} x \sqrt{x + 4}

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

1. ${log}_{x} a + {log}_{x + 4} (\frac{1}{2})$ $log_x a+log_(x+4) (1/2)$
2. ${log}_{x} a + {log}_{a} {(x + 4)}^{\frac{1}{2}}$ $log_x a+log_a (x+4)^(1/2)$
3. ${log}_{a} x + \frac{1}{2} {log}_{a} (x + 4)$ $log_a x+1/2 log_a (x+4)$
4. ${log}_{a} (x + 4) + \frac{1}{2} {log}_{a} x$ $log_a (x+4)+1/2 log_a x$

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Laws of Logarithms

{log}_{b} x y = {log}_{b} x + {log}_{b} y

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

{log}_{b} x^{p} = p {log}_{b} x

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

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^{p}

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

=

$=$

p {log}_{b} x

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

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

$=$ $=$	${log}_{a} x +$ $log_a x+$ $\frac{1}{2}$ $1/2$ ${log}_{a} (x + 4)$ $log_a (x+4)$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

{log}_{a} x + \frac{1}{2} {log}_{a} (x + 4)

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

Question 3 of 3

Expand

{log}_{a} \frac{x (y + z)}{a^{3}}

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

1. ${log}_{a} x + {log}_{a} {(y + z)}^{3}$ $log_a x + log_a (y+z)^3$
2. ${log}_{a} x + {log}_{a} (y + z) + 3$ $log_a x + log_a (y+z) + 3$
3. $3 {log}_{a} x + 3 {log}_{a} (y + z)$ $3log_a x +3 log_a (y+z)$
4. ${log}_{a} x + {log}_{a} (y + z) - 3$ $log_a x + log_a (y+z) - 3$

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Laws of Logarithms

{log}_{b} x y = {log}_{b} x + {log}_{b} y

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

{log}_{b} \frac{x}{y} = {log}_{b} x - {log}_{b} 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^{p} = p {log}_{b} x

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

{log}_{b} b = 1

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

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

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

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

{log}_{a} x + {log}_{a} (y + z) - 3

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

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1. Question

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Great Work!

Laws of Logarithms

2. Question

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Laws of Logarithms

3. Question

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Laws of Logarithms

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