Evaluating Logarithms 2

Question 1 of 5

Solve for

x

$x$

{log}_{9} 27 = x

$log_9 27=x$

$x =$ $x=$ (3/2, 1.5)

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Exponent Form

N = a^{x}

$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$

Logarithmic Form

{log}_{a} N = x

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$

Convert the equation to exponent form by first identifying the components

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$
$\log_{\color{#9a00c7}{9}} \color{#00880a}{27}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$27$ $27$
$a$ $a$	$=$ $=$	$9$ $9$
$x$ $x$	$=$ $=$	$x$ $x$

Substitute the components into the exponent form

$\color{#00880a}{N}$	$=$ $=$	${\color{#9a00c7}{a}}^x$
$\color{#00880a}{27}$	$=$ $=$	${\color{#9a00c7}{9}}^x$

Make sure that only

x

$x$ is on the left side

$27$ $27$	$=$ $=$	$9^{x}$ $9^x$
$27$ $27$	$=$ $=$	${(3^{2})}^{x}$ $(3^2)^x$	$9 = 3^{2}$ $9=3^2$
$3^{3}$ $3^3$	$=$ $=$	$3^{2 x}$ $3^(2x)$	$27 = 3^{3}$ $27=3^3$
$3$ $3$	$=$ $=$	$2 x$ $2x$	Equate the exponents since the bases are equal
$3$ $3$ $\div 2$ $divide 2$	$=$ $=$	$2 x$ $2x$ $\div 2$ $divide 2$	Divide both sides by $2$ $2$

$\frac{3}{2}$ $3/2$	$=$ $=$	$x$ $x$

$x$ $x$	$=$ $=$	$\frac{3}{2}$ $3/2$

x = \frac{3}{2}

$x=3/2$

Question 2 of 5

Solve for

x

$x$

{log}_{8} (\frac{1}{16}) = x

$log_8 (1/16)=x$

$x =$ $x=$ (-4/3)

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Exponent Form

N = a^{x}

$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$

Logarithmic Form

{log}_{a} N = x

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$

Convert the equation to exponent form by first identifying the components

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$

$\log_{\color{#9a00c7}{8}} \color{#00880a}{\frac{1}{16}}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$\frac{1}{16}$ $1/16$

$a$ $a$	$=$ $=$	$8$ $8$
$x$ $x$	$=$ $=$	$x$ $x$

Substitute the components into the exponent form

$\color{#00880a}{N}$	$=$ $=$	${\color{#9a00c7}{a}}^x$

$\color{#00880a}{\frac{1}{16}}$	$=$ $=$	${\color{#9a00c7}{8}}^x$

Make sure that only

x

$x$ is on the left side

$\frac{1}{16}$ $1/16$	$=$ $=$	$8^{x}$ $8^x$

$\frac{1}{16}$ $1/16$	$=$ $=$	${(2^{3})}^{x}$ $(2^3)^x$	$8 = 2^{3}$ $8=2^3$

$\frac{1}{2^{4}}$ $1/(2^4)$	$=$ $=$	$2^{3 x}$ $2^(3x)$	$16 = 2^{4}$ $16=2^4$

$2^{- 4}$ $2^(-4)$	$=$ $=$	$2^{3 x}$ $2^(3x)$	Reciprocate $\frac{1}{2^{4}}$ $1/(2^4)$

$- 4$ $-4$	$=$ $=$	$3 x$ $3x$	Equate the exponents since the bases are equal
$- 4$ $-4$ $\div 3$ $divide3$	$=$ $=$	$3 x$ $3x$ $\div 3$ $divide3$	Divide both sides by $3$ $3$

$- \frac{4}{3}$ $-4/3$	$=$ $=$	$x$ $x$

$x$ $x$	$=$ $=$	$- \frac{4}{3}$ $-4/3$

x = - \frac{4}{3}

$x=-4/3$

Question 3 of 5

Solve for

x

$x$

{log}_{128} 64 = x

$log_128 64=x$

$x =$ $x=$ (6/7)

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Exponent Form

N = a^{x}

$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$

Logarithmic Form

{log}_{a} N = x

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$

Convert the equation to exponent form by first identifying the components

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$
$\log_{\color{#9a00c7}{128}} \color{#00880a}{64}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$64$ $64$
$a$ $a$	$=$ $=$	$128$ $128$
$x$ $x$	$=$ $=$	$x$ $x$

Substitute the components into the exponent form

$\color{#00880a}{N}$	$=$ $=$	${\color{#9a00c7}{a}}^x$

$\color{#00880a}{64}$	$=$ $=$	${\color{#9a00c7}{128}}^x$

Make sure that only

x

$x$ is on the left side

$64$ $64$	$=$ $=$	$128^{x}$ $128^x$
$64$ $64$	$=$ $=$	${(2^{7})}^{x}$ $(2^7)^x$	$128 = 2^{7}$ $128=2^7$
$2^{6}$ $2^6$	$=$ $=$	$2^{7 x}$ $2^(7x)$	$64 = 2^{6}$ $64=2^6$
$6$ $6$	$=$ $=$	$7 x$ $7x$	Equate the exponents since the bases are equal
$6$ $6$ $\div 7$ $divide 7$	$=$ $=$	$7 x$ $7x$ $\div 7$ $divide 7$	Divide both sides by $7$ $7$

$\frac{6}{7}$ $6/7$	$=$ $=$	$x$ $x$

$x$ $x$	$=$ $=$	$\frac{6}{7}$ $6/7$

x = \frac{6}{7}

$x=6/7$

Question 4 of 5

Solve for

{log}_{a} 6

$log_a 6$ , given that:

{log}_{a} 2 = 0.3010

$log_a 2=0.3010$

{log}_{a} 3 = 0.4771

$log_a 3=0.4771$

Round your answer to 4 decimal places

${log}_{a} 6 =$ $log_a 6=$ (0.7781, .7781)

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Laws of Logarithms

{log}_{b} x y = {log}_{b} x + {log}_{b} y

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

Expand the given logarithmic expression

	$\log_a 6$
$=$ $=$	$\log_a (3)(2)$	$6 = (3) (2)$ $6=(3)(2)$
$=$ $=$	$\log_\color{#9a00c7}{a} \color{#00880A}{(3)}\color{#e65021}{(2)}$
$=$ $=$	$\log_\color{#9a00c7}{a} \color{#00880A}{3}+\log_\color{#9a00c7}{a} \color{#e65021}{2}$	${log}_{b} x y = {log}_{b} x + {log}_{b} y$ $log_b xy=log_b x+log_b y$

Substitute the given values

$\log_a 2$	$=$ $=$	$0.3010$
$\log_a 3$	$=$ $=$	$0.4771$

		${log}_{a} 3 + {log}_{a} 2$ $log_a 3+log_a 2$
	$=$ $=$	$0.4771 + 0.3010$ $0.4771+0.3010$
	$=$ $=$	$0.7781$ $0.7781$

0.7781

$0.7781$

Question 5 of 5

Solve for

{log}_{a} (\frac{4}{9})

$log_a (4/9)$ , given that:

{log}_{a} 2 = 0.3010

$log_a 2=0.3010$

{log}_{a} 3 = 0.4771

$log_a 3=0.4771$

Round your answer to 4 decimal places

${log}_{a} (\frac{4}{9}) =$ $log_a (4/9)=$ (-0.3522)

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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

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

Expand the given logarithmic expression

	$\log_a \frac{4}{9}$

$=$ $=$	$\log_a \frac{2^2}{9}$	$4 = 2^{2}$ $4=2^2$

$=$ $=$	$\log_a \frac{2^2}{3^3}$	$3 = 3^{2}$ $3=3^2$

$=$ $=$	$\log_\color{#9a00c7}{a} {\frac{\color{#00880A}{2^2}}{\color{#e65021}{3^2}}}$

$=$ $=$	$\log_\color{#9a00c7}{a} \color{#00880A}{2^2}-\log_\color{#9a00c7}{a} \color{#e65021}{3^2}$	$log_b \frac{x}{y}=log_b x-\log_b y$
$=$ $=$	$\color{#004ec4}{2}\log_{a} 2-\color{#004ec4}{2}\log_{a} 3$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

Substitute the given values

$\log_a 2$	$=$ $=$	$0.3010$
$\log_a 3$	$=$ $=$	$0.4771$

		$2 {log}_{a} 2 - 2 {log}_{a} 3$ $2\log_a 2-2\log_a 3$
	$=$ $=$	$2 (0.3010) - 2 (0.4771)$ $2(0.3010)-2(0.4771)$
	$=$ $=$	$0.60206 - 0.95424$ $0.60206-0.95424$
	$=$ $=$	$- 0.3522$ $-0.3522$

- 0.3522

$-0.3522$

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Quiz summary

Information

1. Question

Hint

Excellent!

Exponent Form

Logarithmic Form

2. Question

Hint

Nice Job!

Exponent Form

Logarithmic Form

3. Question

Hint

Well Done!

Exponent Form

Logarithmic Form

4. Question

Hint

Well Done!

Laws of Logarithms

5. Question

Hint

Excellent!

Laws of Logarithms

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