Simplifying Log Expressions 3

Question 1 of 5

Simplify and solve

{log}_{8} 4 - {log}_{8} 32 + {log}_{8} 64

$log_8 4-log_8 32+log_8 64$

(1)

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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} b = 1

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

Compress the expression into one logarithm

	$\log_\color{#9a00c7}{8} \color{#00880A}{4}-\log_8 32+\log_\color{#9a00c7}{8} \color{#e65021}{64}$
$=$ $=$	$\log_\color{#9a00c7}{8} \color{#00880A}{(4)}\color{#e65021}{(64)}-\log_2 18$	${log}_{b} x y = {log}_{b} x + {log}_{b} y$ $log_b xy=log_b x+log_b y$
$=$ $=$	$\log_8 256-\log_8 32$
$=$ $=$	$\log_\color{#9a00c7}{8} \color{#00880A}{256}-\log_\color{#9a00c7}{8} \color{#e65021}{32}$

$=$ $=$	$\log_\color{#9a00c7}{8} \frac{\color{#00880A}{256}}{\color{#e65021}{32}}$	$log_b \frac{x}{y}=log_b x-\log_b y$

$=$ $=$	${log}_{8} 8$ $log_8 8$
$=$ $=$	$1$ $1$	${log}_{b} b = 1$ $log_b b=1$

1

$1$

Question 2 of 5

Simplify and solve

{log}_{10} 250 + {log}_{10} 16 - {log}_{10} 4

$log_10 250+log_10 16-log_10 4$

(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

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

Compress the expression into one logarithm

	$\log_\color{#9a00c7}{10} \color{#00880A}{250}+\log_\color{#9a00c7}{10} \color{#e65021}{16}-\log_{10} 4$
$=$ $=$	$\log_\color{#9a00c7}{10} \color{#00880A}{(250)}\color{#e65021}{(16)}-\log_{10} 4$	${log}_{b} x y = {log}_{b} x + {log}_{b} y$ $log_b xy=log_b x+log_b y$
$=$ $=$	$\log_{10} 4000-\log_{10} 4$
$=$ $=$	$\log_\color{#9a00c7}{10} \color{#00880A}{4000}-\log_\color{#9a00c7}{10} \color{#e65021}{4}$

$=$ $=$	$\log_\color{#9a00c7}{10} \frac{\color{#00880A}{4000}}{\color{#e65021}{4}}$	$log_b \frac{x}{y}=log_b x-\log_b y$

$=$ $=$	${log}_{10} 1000$ $log_(10) 1000$

Simplify the logarithm

	${log}_{10} 1000$ $log_(10) 1000$
$=$ $=$	$\log_{10} \color{#CC0000}{10^3}$	$1000=10^3$
$=$ $=$	$\color{#004ec4}{3}\log_{10} 10$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$
$=$ $=$	$3(\color{#9a00c7}{1})$	${log}_{b} b = 1$ $log_b b=1$
$=$ $=$	$3$ $3$

3

$3$

Question 3 of 5

Simplify and solve

3 {log}_{2} 4 + \frac{1}{2} {log}_{2} 81 - {log}_{2} 18

$3log_2 4+1/2log_2 81-log_2 18$

(5)

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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

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

Remove the coefficients from the logarithms so they can be combined

First term:

	$3\log_2 4+\frac{1}{2}\log_2 81-\log_2 18$

$=$ $=$	$\log_2 4^\color{#004ec4}{3}+\frac{1}{2}\log_2 81-\log_2 18$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

$=$ $=$	$\log_2 64+\frac{1}{2}\log_2 81-\log_2 18$

Second term:

$=$ $=$	$\log_2 64+\frac{1}{2}\log_2 81-\log_2 18$

$=$ $=$	$\log_2 64+\log_2 81^\color{#004ec4}{\frac{1}{2}}-\log_2 18$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

$=$ $=$	$\log_2 64+\log_2 \color{#CC0000}{\sqrt{81}}-\log_2 18$	Change the exponent into a surd
$=$ $=$	${log}_{2} 64 + {log}_{2} 9 - {log}_{2} 18$ $log_2 64+log_2 9-log_2 18$

Compress the expression into one logarithm

	$\log_\color{#9a00c7}{2} \color{#00880A}{64}+\log_\color{#9a00c7}{2} \color{#e65021}{9}-\log_2 18$
$=$	$\log_\color{#9a00c7}{2} \color{#00880A}{(64)}\color{#e65021}{(9)}-\log_2 18$	$log_b xy=log_b x+log_b y$
$=$	$log_2 576-log_2 18$
$=$	$\log_\color{#9a00c7}{2} \color{#00880A}{576}-\log_\color{#9a00c7}{2} \color{#e65021}{18}$

$=$	$\log_\color{#9a00c7}{2} \frac{\color{#00880A}{576}}{\color{#e65021}{18}}$	$log_b \frac{x}{y}=log_b x-\log_b y$

$=$	$log_2 32$

Simplify the logarithm

	$log_2 32$
$=$	$\log_2 \color{#CC0000}{2^5}$	$32=2^5$
$=$	$\color{#004ec4}{5}\log_2 2$	$log_b x^p=p log_b x$
$=$	$5($ $1$ $)$	$log_b b=1$
$=$	$5$

$5$

Question 4 of 5

Simplify and solve

$(log_a x^3-log_a x^2)/log_a sqrtx$

(2)

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Laws of Logarithms

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

Compress the expression into one logarithm

	$\frac{\log_a x^3-\log_a x^2}{\log_a \sqrt{x}}$

$=$	$\frac{\log_a x^3-\log_a x^2}{\log_a x^\color{#CC0000}{\frac{1}{2}}}$	Change the surd into an exponent

$=$	$\frac{\color{#004ec4}{3}\log_a x-\color{#004ec4}{2}\log_a x}{\color{#004ec4}{\frac{1}{2}}\log_a x}$	$log_b x^p=p log_b x$

$=$	$\frac{1\log_a x}{\frac{1}{2}\log_a x}$

$=$	$\frac{1}{\frac{1}{2}}$	$(log_a x)/(log_a x)=1$

$=$	$2$

$2$

Question 5 of 5

Simplify and solve

$(12log_a sqrta-log_a (1/(a^2)))/log_a a^2$

(4)

Incorrect

Laws of Logarithms

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

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

Compress the expression into one logarithm

	$\frac{12\log_a \sqrt{a}-\log_a \frac{1}{a^2}}{\log_a a^2}$

$=$	$\frac{12\log_a \sqrt{a}-\log_a \color{#CC0000}{a^{-2}}}{\log_a a^2}$	Reciprocate $1/a^2$

$=$	$\frac{\color{#004ec4}{12}\log_a a^\color{#CC0000}{\frac{1}{2}}-\log_a a^{-2}}{\log_a a^2}$	Change the surd into an exponent

$=$	$\frac{\log_a a^{(\color{#004ec4}{12})(\frac{1}{2})}-\log_a a^{-2}}{\log_a a^2}$	$log_b x^p=p log_b x$

$=$	$\frac{\log_a a^6-\log_a a^{-2}}{\log_a a^2}$

$=$	$\frac{\log_\color{#9a00c7}{a} \color{#00880A}{a^6}-\log_\color{#9a00c7}{a} \color{#e65021}{a^{-2}}}{\log_a a^2}$

$=$	$\frac{\log_\color{#9a00c7}{a} \frac{\color{#00880A}{a^6}}{\color{#e65021}{a^{-2}}}}{\log_a a^2}$	$log_b \frac{x}{y}=log_b x-\log_b y$

$=$	$\frac{\log_a a^8}{\log_a a^2}$

$=$	$\frac{\color{#004ec4}{8}\log_a a}{\color{#004ec4}{2}\log_a a}$	$log_b x^p=p log_b x$

$=$	$\frac{8}{2}$	$(log_a x)/(log_a x)=1$

$=$	$4$

$4$

Simplifying Log Expressions 3

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