Simplifying Log Expressions 2

Question 1 of 4

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$log_5 125-3log_5 25$

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Laws of Logarithms

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

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

Use the following laws to simplify the logarithmic expression

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

	$log_5 125-3log_5 25$
$=$	$log_5$ $5^3$ $-3log_5 25$	$125=5^3$
$=$	$log_5 5^3-3log_5$ $5^2$	$25=5^2$
$=$	$3$ $log_5 5-3($ $2$ $)log_5 5$	$log_b x^p=p log_b x$
$=$	$3($ $1$ $)-6($ $1$ $)$	$log_b b=1$
$=$	$3-6$
$=$	$-3$

$-3$

Question 2 of 4

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$3log_10 2+log_10 12.5$

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Laws of Logarithms

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

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

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

Remove the coefficients from the logarithms so they can be combined

	$3\log_{10} 2+\log_{10} 12.5$
$=$	$\log_{10} 2^\color{#004ec4}{3}+\log_{10} 12.5$	$log_b x^p=p log_b x$
$=$	$\log_{10} 8+\log_{10} 12.5$

Next, compare the given expression to one of the laws of logarithms and identify corresponding components

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$	$+$	$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$
$\log_{\color{#9a00c7}{10}} \color{#00880A}{8}$	$+$	$\log_{\color{#9a00c7}{10}} \color{#e65021}{12.5}$

$b$	$=$	$10$
$x$	$=$	$8$
$y$	$=$	$12.5$

Substitute the components into the law of logarithms

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$	$=$	$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$
$\log_{\color{#9a00c7}{10}} \color{#00880A}{8} + \log_{\color{#9a00c7}{10}} \color{#e65021}{12.5}$	$=$	$\log_{\color{#9a00c7}{10}} {\color{#00880A}{(8)}\color{#e65021}{(12.5)}}$
	$=$	$log_10 100$

Simplify the logarithm

	$log_10 100$
$=$	$log_10$ $10^2$	$100=10^2$
$=$	$\color{#004ec4}{2}\log_{10} 10$	$log_b x^p=p log_b x$
$=$	$2($ $1$ $)$	$log_b b=1$
$=$	$2$

$2$

Question 3 of 4

Solve

$2log_3 root(3)(3)+log_3 sqrt27$

1. $2 1/9$
2. $2 1/6$
3. $5/6$
4. $4/9$

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Laws of Logarithms

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

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

Use the following laws to simplify the logarithmic expression

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

	$2log_3 root(3)(3)+log_3 sqrt27$
$=$	$2\log_3 \sqrt[3]{3}+\log_3 \sqrt{\color{#CC0000}{3^3}}$	$27=3^3$
$=$	$2\log_3 3^\color{#CC0000}{\frac{1}{3}}+\log_3 3^\color{#CC0000}{\frac{3}{2}}$	Change the surds into exponents

$=$	$1/3$ $(2)log_3 3+$ $3/2$ $log_3 3$	$log_b x^p=p log_b x$

$=$	$2/3log_3 3+3/2log_3 3$

$=$	$2/3 xx$ $1$ $+3/2 xx$ $1$	$log_b b=1$

$=$	$2/3+3/2$

$=$	$4/6+9/6$	Find the common denominator

$=$	$13/6$

$=$	$2 1/6$

$2 1/6$

Question 4 of 4

Solve

$log_a ((x^3)/(x+2))+log_a (x+2)$

1. $log_a 3x$
2. $3 log_3 x$
3. $a log_3 x$
4. $3 log_a x$

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Laws of Logarithms

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

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

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

First, compare the given expression to one of the laws of logarithms and identify corresponding components

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

$\log_{\color{#9a00c7}{a}} \color{#00880A}{\frac{x^3}{x+2}}$	$+$	$\log_{\color{#9a00c7}{a}} (\color{#e65021}{x+2})$

$b$	$=$	$a$

$x$	$=$	$(x^3)/(x+2)$

$y$	$=$	$x+2$

Substitute the components into the law of logarithms

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

$\log_{\color{#9a00c7}{a}} \color{#00880A}{\frac{x^3}{x+2}} + \log_{\color{#9a00c7}{a}} (\color{#e65021}{x+2})$	$=$	$\log_{\color{#9a00c7}{a}} {\color{#00880A}{\frac{x^3}{x+2}}\times{\color{#e65021}{x+2}}}$

	$=$	$log_a x^3$	$(x+2)/(x+2)=1$

Simplify the logarithm

$log_a x^3$

$=$

$3$

$log_a x$

$log_b x^p=p log_b x$

$3 log_a x$

Simplifying Log Expressions 2

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Quiz summary

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1. Question

Hint

Correct!

Laws of Logarithms

2. Question

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Laws of Logarithms

3. Question

Hint

Great Work!

Laws of Logarithms

4. Question

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Great Work!

Laws of Logarithms

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