Logarithmic Equations 2

Question 1 of 5

Solve for

x

$x$

{log}_{2} x = {log}_{2} y + 4 {log}_{2} z

$log_2 x=log_2 y+4log_2 z$

1. $x = y z^{4}$ $x = yz^4$
2. $x = 4 y z$ $x = 4yz$
3. $x = {(y z)}^{4}$ $x = (yz)^4$
4. $x = y z$ $x = yz$

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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

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

Remove the coefficient from the second term

$\log_{2} x$	$=$ $=$	$\log_{2} y+4\log_{2} z$
$\log_{2} x$	$=$ $=$	$\log_{2} y+\log_{2} z^\color{#004ec4}{4}$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

Contract the right side

$\log_{2} x$	$=$	$\log_\color{#9a00c7}{2} \color{#00880A}{y}+\log_\color{#9a00c7}{2} \color{#e65021}{z^4}$
$\log_{2} x$	$=$	$\log_\color{#9a00c7}{2} {\color{#00880A}{y}}{\color{#e65021}{z^4}}$	$log_b xy=log_b x+log_b y$

Since the bases of both sides are the same, the logarithm can be dropped

$\log_{2} \color{#00880A}{x}$	$=$	$\log_{2} \color{#00880A}{yz^4}$
$\color{#00880A}{x}$	$=$	$\color{#00880A}{yz^4}$

$x=yz^4$

Question 2 of 5

Solve for

$a$

$log_10 a=3log_10 x-1$

1. $a=(x^3)/10$
2. $a=1/(x^3)$
3. $a=x^3$
4. $a=(3x)/10$

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Laws of Logarithms

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

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

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

Remove the coefficient from the second term

$\log_{10} a$	$=$	$3\log_{10} x-1$
$\log_{10} a$	$=$	$\log_{10} x^\color{#004ec4}{3}-1$	$log_b x^p=p log_b x$

Transform the constant (third term) into a logarithmic term

$\log_{10} a$	$=$	$\log_{10} x^3-1$
$\log_{10} a$	$=$	$\log_{10} x^3-\log_\color{#9a00c7}{10} \color{#9a00c7}{10}$	$1=log_{10} 10$

Contract the right side

$\log_{10} a$	$=$	$\log_\color{#9a00c7}{10} \color{#00880A}{x^3}-\log_\color{#9a00c7}{10} \color{#e65021}{10}$

$\log_{10} a$	$=$	$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{x^3}}{\color{#e65021}{10}}}$	$log_b \frac{x}{y}=log_b x-\log_b y$

Since the bases of both sides are the same, the logarithm can be dropped

$\log_{10} \color{#00880A}{a}$	$=$	$\log_{10} \color{#00880A}{\frac{x^3}{10}}$

$\color{#00880A}{a}$	$=$	$\color{#00880A}{\frac{x^3}{10}}$

$a=(x^3)/10$

Question 3 of 5

Solve for

$x$

$log_a x=3log_a y+1$

1. $x = y^3a$
2. $x = ya^3$
3. $x = y^3/a$
4. $x = 3ay$

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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 coefficient from the second term

$\log_{a} x$	$=$	$3\log_{a} y+1$
$\log_{a} x$	$=$	$\log_{a} y^\color{#004ec4}{3}+1$	$log_b x^p=p log_b x$

Transform the constant (third term) into a logarithmic term

$\log_{a} x$	$=$	$\log_{a} y^3+1$
$\log_{a} x$	$=$	$\log_{a} y^3+\log_\color{#9a00c7}{a} \color{#9a00c7}{a}$	$1=log_{a} a$

Contract the right side

$\log_{a} x$	$=$	$\log_\color{#9a00c7}{a} \color{#00880A}{y^3}+\log_\color{#9a00c7}{a} \color{#e65021}{a}$
$\log_{a} x$	$=$	$\log_\color{#9a00c7}{a} {\color{#00880A}{y^3}}{\color{#e65021}{a}}$	$log_b xy=log_b x+log_b y$

Since the bases of both sides are the same, the logarithm can be dropped

$\log_{a} \color{#00880A}{x}$	$=$	$\log_{a} \color{#00880A}{y^3a}$
$\color{#00880A}{x}$	$=$	$\color{#00880A}{y^3a}$

$x=y^3a$

Question 4 of 5

Solve for

$a$

$log_10 a=2-log_10 x$

1. $a= x/100$
2. $a= 100x$
3. $a= 100/x$
4. $a= 100^x$

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Laws of Logarithms

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

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

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

Add a logarithmic term to the constant (second term)

$\log_{10} a$	$=$	$2-\log_{10} x$
$\log_{10} a$	$=$	$2\log_\color{#9a00c7}{10} \color{#9a00c7}{10}-\log_{10} x$	$1=log_{10} 10$

Remove the coefficient from the second term

$\log_{10} a$	$=$	$2\log_{10} 10-\log_{10} x$
$\log_{10} a$	$=$	$\log_{10} 10^\color{#004ec4}{2}-\log_{10} x$	$log_b x^p=p log_b x$

Contract the right side

$\log_{10} a$	$=$	$\log_\color{#9a00c7}{10} \color{#00880A}{10^2}-\log_\color{#9a00c7}{10} \color{#e65021}{x}$

$\log_{10} a$	$=$	$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{10^2}}{\color{#e65021}{x}}}$	$log_b \frac{x}{y}=log_b x-\log_b y$

Since the bases of both sides are the same, the logarithm can be dropped

$\log_{10} \color{#00880A}{a}$	$=$	$\log_{10} \color{#00880A}{\frac{10^2}{x}}$

$\color{#00880A}{a}$	$=$	$\color{#00880A}{\frac{100}{x}}$

$a=(100)/x$

Question 5 of 5

Solve for

$x$

$log_10 2+2log_10 x-log_10 50=0$

$x=$ (5)

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Laws of Logarithms

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

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

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

$\log_{\color{#9a00c7}{b}} 1=0$

Remove the coefficient from the second term

$\log_{10} 2+2\log_{10} x-\log_{10} 50$	$=$	$0$
$\log_{10} 2+\log_{10} x^\color{#004ec4}{2}-\log_{10} 50$	$=$	$0$	$log_b x^p=p log_b x$

Transform the constant (fourth term) into a logarithmic term

$\log_{10} 2+\log_{10} x^2-\log_{10} 50$	$=$	$0$
$\log_{10} 2+\log_{10} x^2-\log_{10} 50$	$=$	$\log_\color{#9a00c7}{10} 1$	$0=log_{10} 1$

Contract the left side

$\log_\color{#9a00c7}{10} \color{#00880A}{2}+\log_\color{#9a00c7}{10} \color{#e65021}{x^2}-\log_{10} 50$	$=$	$\log_{10} 1$
$\log_\color{#9a00c7}{10} \color{#00880A}{2}\color{#e65021}{x^2}-\log_{10} 50$	$=$	$\log_{10} 1$	$log_b xy=log_b x+log_b y$
$\log_\color{#9a00c7}{10} \color{#00880A}{2x^2}-\log_\color{#9a00c7}{10} \color{#e65021}{50}$	$=$	$\log_{10} 1$

$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{2x^2}}{\color{#e65021}{50}}}$	$=$	$\log_{10} 1$	$log_b \frac{x}{y}=log_b x-\log_b y$

Since the bases of both sides are the same, the logarithm can be dropped

$\log_{10} \color{#00880A}{\frac{2x^2}{50}}$	$=$	$\log_{10} \color{#00880A}{1}$

$\color{#00880A}{\frac{2x^2}{50}}$	$=$	$\frac{1}{1}$

$2x^2$	$=$	$50$	Cross multiply
$2x^2$ $divide2$	$=$	$50$ $divide2$	Divide both sides by $2$
$sqrt(x^2)$	$=$	$sqrt25$	Get the square root of both sides
$x$	$=$	$5$

$x=5$

Logarithmic Equations 2

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