Logarithmic Equations 1

Topics

Algebra 2

Logarithms

Logarithmic Equations

Logarithmic Equations 1

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Question 1 of 4

1. Question
Solve for $N$ $N$

${log}_{5} N = - 3$ $log_5 N=-3$
- $N =$ $N=$ (1/125)
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Exponent Form

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

Logarithmic Form

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

Convert the equation to exponent form by first identifying the components

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

$\log_{\color{#9a00c7}{5}} \color{#00880a}{N}$ $=$ $=$ $-3$

$N$ $N$ $=$ $=$ $N$ $N$

$a$ $a$ $=$ $=$ $5$ $5$

$x$ $x$ $=$ $=$ $- 3$ $-3$

Substitute the components into the exponent form

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

$\color{#00880a}{N}$ $=$ $=$ ${\color{#9a00c7}{5}}^{-3}$

Solve for the value of $N$ $N$

$N$ $N$ $=$ $=$ $5^{- 3}$ $5^(-3)$

$N$ $N$ $=$ $=$ $\frac{1}{5^{3}}$ $1/(5^3)$ Reciprocate $5^{- 3}$ $5^(-3)$

$N$ $N$ $=$ $=$ $\frac{1}{125}$ $1/125$

$N = \frac{1}{125}$ $N=1/125$
Question 2 of 4

2. Question
Solve for $N$ $N$

${log}_{3} N = - 4$ $log_3 N=-4$
- $N =$ $N=$ (1/81)
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Exponent Form

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

Logarithmic Form

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

Convert the equation to exponent form by first identifying the components

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

$\log_{\color{#9a00c7}{3}} \color{#00880a}{N}$ $=$ $=$ $-4$

$N$ $N$ $=$ $=$ $N$ $N$

$a$ $a$ $=$ $=$ $3$ $3$

$x$ $x$ $=$ $=$ $- 4$ $-4$

Substitute the components into the exponent form

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

$\color{#00880a}{N}$ $=$ $=$ ${\color{#9a00c7}{3}}^{-4}$

Solve for the value of $N$ $N$

$N$ $N$ $=$ $=$ $3^{- 4}$ $3^(-4)$

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

$N$ $N$ $=$ $=$ $\frac{1}{81}$ $1/81$

$N = \frac{1}{81}$ $N=1/81$
Question 3 of 4

3. Question
Solve for $a$ $a$

${log}_{a} 27 = 3$ $log_a 27=3$
- $a =$ $a=$ (3)
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Exponent Form

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

Logarithmic Form

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

Convert the equation to exponent form by first identifying the components

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

$\log_{\color{#9a00c7}{a}} \color{#00880a}{27}$ $=$ $=$ $3$

$N$ $N$ $=$ $=$ $27$ $27$

$a$ $a$ $=$ $=$ $a$ $a$

$x$ $x$ $=$ $=$ $3$ $3$

Substitute the components into the exponent form

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

$\color{#00880a}{27}$ $=$ $=$ ${\color{#9a00c7}{a}}^{3}$

Solve for the value of $a$ $a$

$27$ $27$ $=$ $=$ $a^{3}$ $a^3$

$\sqrt[3]{27}$ $root (3)(27)$ $=$ $=$ $\sqrt[3]{a^{3}}$ $root (3)(a^3)$ Find the cube root of both sides

$3$ $3$ $=$ $=$ $a$ $a$

$a$ $a$ $=$ $=$ $3$ $3$

$a = 3$ $a=3$

Question 4 of 4

Solve for

x

$x$

{log}_{b} x = 3 {log}_{b} 2 + {log}_{b} 4

$log_b x=3log_b 2+log_b 4$

$x =$ $x=$ (32)

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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

$\log_{b} x$	$=$ $=$	$3\log_{b} 2+\log_{b} 4$
$\log_{b} x$	$=$ $=$	$\log_{b} 2^\color{#004ec4}{3}+\log_{b} 4$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$
$\log_{b} x$	$=$ $=$	$\log_{b} 8+\log_{b} 4$

Contract the right side

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

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

$\log_{b} \color{#00880A}{x}$	$=$ $=$	$\log_{b} \color{#00880A}{32}$
$\color{#00880A}{x}$	$=$ $=$	$\color{#00880A}{32}$

x = 32

$x=32$

Logarithmic Equations 1

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Quiz summary

Information

1. Question

Hint

Great Work!

Exponent Form

Logarithmic Form

2. Question

Hint

Correct!

Exponent Form

Logarithmic Form

3. Question

Hint

Keep Going!

Exponent Form

Logarithmic Form

4. Question

Hint

Great Work!

Laws of Logarithms

Quizzes

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$
$\log_{\color{#9a00c7}{5}} \color{#00880a}{N}$	$=$ $=$	$-3$

$N$ $N$	$=$ $=$	$N$ $N$
$a$ $a$	$=$ $=$	$5$ $5$
$x$ $x$	$=$ $=$	$- 3$ $-3$

$\color{#00880a}{N}$	$=$ $=$	${\color{#9a00c7}{a}}^x$
$\color{#00880a}{N}$	$=$ $=$	${\color{#9a00c7}{5}}^{-3}$

$N$ $N$	$=$ $=$	$5^{- 3}$ $5^(-3)$

$N$ $N$	$=$ $=$	$\frac{1}{5^{3}}$ $1/(5^3)$	Reciprocate $5^{- 3}$ $5^(-3)$

$N$ $N$	$=$ $=$	$\frac{1}{125}$ $1/125$

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$
$\log_{\color{#9a00c7}{3}} \color{#00880a}{N}$	$=$ $=$	$-4$

$N$ $N$	$=$ $=$	$3^{- 4}$ $3^(-4)$

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

$N$ $N$	$=$ $=$	$\frac{1}{81}$ $1/81$

$N$ $N$	$=$ $=$	$27$ $27$
$a$ $a$	$=$ $=$	$a$ $a$
$x$ $x$	$=$ $=$	$3$ $3$

$27$ $27$	$=$ $=$	$a^{3}$ $a^3$
$\sqrt[3]{27}$ $root (3)(27)$	$=$ $=$	$\sqrt[3]{a^{3}}$ $root (3)(a^3)$	Find the cube root of both sides
$3$ $3$	$=$ $=$	$a$ $a$
$a$ $a$	$=$ $=$	$3$ $3$