Simplifying Log Expressions 1

Question 1 of 4

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\log_{6} 12 + \log_{6} 3

$log_6 12+log_6 3$

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Laws of Logarithms

\log_{b} x y = \log_{b} x + \log_{b} y

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

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

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$	$+$ $+$	$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$
$\log_{\color{#9a00c7}{6}} \color{#00880A}{12}$	$+$ $+$	$\log_{\color{#9a00c7}{6}} \color{#e65021}{3}$

$b$ $b$	$=$ $=$	$6$ $6$
$x$ $x$	$=$ $=$	$12$ $12$
$y$ $y$	$=$ $=$	$3$ $3$

Substitute the components into the law of logarithms

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$	$=$ $=$	$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$
$\log_{\color{#9a00c7}{6}} \color{#00880A}{12} + \log_{\color{#9a00c7}{6}} \color{#e65021}{3}$	$=$ $=$	$\log_{\color{#9a00c7}{6}} {\color{#00880A}{(12)}\color{#e65021}{(3)}}$
	$=$ $=$	$\log_{6} 36$ $log_6 36$

Let the simplified logarithm be equal to

a

$a$

$a$ $a$	$=$ $=$	$\log_{6} 36$ $log_6 36$
$6^{a}$ $6^a$	$=$ $=$	$36$ $36$	Convert to exponent form
$6^{a}$ $6^a$	$=$ $=$	$6^{2}$ $6^2$	$36 = 6^{2}$ $36=6^2$
$a$ $a$	$=$ $=$	$2$ $2$	Equate the exponents since the bases are equal

2

$2$

Question 2 of 4

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\log_{2} 48 - \log_{2} 3

$log_2 48-log_2 3$

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Laws of Logarithms

\log_{b} \frac{x}{y} = \log_{b} x - \log_{b} y

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

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

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$	$-$ $-$	$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$
$\log_{\color{#9a00c7}{2}} \color{#00880A}{48}$	$-$ $-$	$\log_{\color{#9a00c7}{2}} \color{#e65021}{3}$

$b$ $b$	$=$ $=$	$2$ $2$
$x$ $x$	$=$ $=$	$48$ $48$
$y$ $y$	$=$ $=$	$3$ $3$

Substitute the components into the law of logarithms

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

$\log_{\color{#9a00c7}{2}} \color{#00880A}{48}-\log_{\color{#9a00c7}{2}} \color{#e65021}{3}$	$=$ $=$	$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{48}}{\color{#e65021}{3}}$

	$=$ $=$	$\log_{2} 16$ $log_2 16$

Let the simplified logarithm be equal to

a

$a$

$a$ $a$	$=$ $=$	$\log_{2} 16$ $log_2 16$
$2^{a}$ $2^a$	$=$ $=$	$16$ $16$	Convert to exponent form
$2^{a}$ $2^a$	$=$ $=$	$2^{4}$ $2^4$	$16 = 2^{4}$ $16=2^4$
$a$ $a$	$=$ $=$	$4$ $4$	Equate the exponents since the bases are equal

4

$4$

Question 3 of 4

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\log_{10} 25 + \log_{10} 4

$log_10 25+log_10 4$

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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

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

\log_{b} b = 1

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

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

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$	$+$ $+$	$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$
$\log_{\color{#9a00c7}{10}} \color{#00880A}{25}$	$+$ $+$	$\log_{\color{#9a00c7}{10}} \color{#e65021}{4}$

$b$ $b$	$=$ $=$	$10$ $10$
$x$ $x$	$=$ $=$	$25$ $25$
$y$ $y$	$=$ $=$	$4$ $4$

Substitute the components into the law of logarithms

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$	$=$ $=$	$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$
$\log_{\color{#9a00c7}{10}} \color{#00880A}{25} + \log_{\color{#9a00c7}{10}} \color{#e65021}{4}$	$=$ $=$	$\log_{\color{#9a00c7}{10}} {\color{#00880A}{(25)}\color{#e65021}{(4)}}$
	$=$ $=$	$\log_{10} 100$ $log_10 100$

Simplify the logarithm

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

2

$2$

Question 4 of 4

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\log_{5} 1000 - \log_{5} 8

$log_5 1000-log_5 8$

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Laws of Logarithms

\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

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

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

$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$	$-$ $-$	$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$
$\log_{\color{#9a00c7}{5}} \color{#00880A}{1000}$	$-$ $-$	$\log_{\color{#9a00c7}{5}} \color{#e65021}{8}$

$b$ $b$	$=$ $=$	$5$ $5$
$x$ $x$	$=$ $=$	$1000$ $1000$
$y$ $y$	$=$ $=$	$8$ $8$

Substitute the components into the law of logarithms

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

$\log_{\color{#9a00c7}{5}} \color{#00880A}{1000}-\log_{\color{#9a00c7}{5}} \color{#e65021}{8}$	$=$ $=$	$\log_{\color{#9a00c7}{5}} \frac{\color{#00880A}{1000}}{\color{#e65021}{8}}$

	$=$ $=$	$\log_{5} 125$ $log_5 125$

Simplify the logarithm

	$\log_{5} 125$ $log_5 125$
$=$ $=$	$\log_{5}$ $log_5$ $5^{3}$ $5^3$	$125 = 5^{3}$ $125=5^3$
$=$ $=$	$3$ $3$ $\log_{5} 5$ $log_5 5$	$\log_{b} x^{p} = p \log_{b} x$ $log_b x^p=p log_b x$
$=$ $=$	$3 ($ $3($ $1$ $1$ $)$ $)$	$\log_{b} b = 1$ $log_b b=1$
$=$ $=$	$3$ $3$

3

$3$

Simplifying Log Expressions 1

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