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. $\log_{a} x + \log_{a} y - \log_{a} z$ $log_a x + log_a y - log_a z$

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

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

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

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. $3 \log_{a} x + 3 \log_{a} (y + z)$ $3log_a x +3 log_a (y+z)$
3. $\log_{a} x + \log_{a} (y + z) + 3$ $log_a x + log_a (y+z) + 3$
4. $\log_{a} x + \log_{a} (y + z) - 3$ $log_a x + log_a (y+z) - 3$

Incorrect

Need Text

Play

Remaining Time -0:00

Stream TypeLIVE

Loaded: 0%

Progress: 0%

0:00

Fullscreen

00:00

Mute

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

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

Try VividMath Premium to unlock full access

Quiz summary

Information

1. Question

Hint

Great Work!

Laws of Logarithms

2. Question

Hint

Correct!

Laws of Logarithms

3. Question

Hint

Keep Going!

Laws of Logarithms

Quizzes