Evaluating Logarithms 1

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Year 10

Logarithms

Evaluating Logarithms

Evaluating Logarithms 1

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Question 1 of 4

1. Question
Solve for $x$ $x$

$x = \log_{4} 64$ $x=log_4 64$
- $x =$ $x=$ (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

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

$\log_{\color{#9a00c7}{4}} \color{#00880a}{64}$ $=$ $=$ $x$

$N$ $N$ $=$ $=$ $64$ $64$

$a$ $a$ $=$ $=$ $4$ $4$

$x$ $x$ $=$ $=$ $x$ $x$

Substitute the components into the exponent form

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

$\color{#00880a}{64}$ $=$ $=$ ${\color{#9a00c7}{4}}^x$

Make sure that only $x$ $x$ is on the left side

$64$ $64$ $=$ $=$ $4^{x}$ $4^x$

$4^{3}$ $4^3$ $=$ $=$ $4^{x}$ $4^x$ $64 = 4^{3}$ $64=4^3$

$3$ $3$ $=$ $=$ $x$ $x$ Equate the exponents since the bases are equal

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

$x = 3$ $x=3$
Question 2 of 4

2. Question
Solve for $x$ $x$

$x = \log_{8} 1$ $x=log_8 1$
- $x =$ $x=$ (0)
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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

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

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

$N$ $N$ $=$ $=$ $1$ $1$

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

$x$ $x$ $=$ $=$ $x$ $x$

Substitute the components into the exponent form

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

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

Make sure that only $x$ $x$ is on the left side

$1$ $1$ $=$ $=$ $8^{x}$ $8^x$

To make the equation correct, we need to raise $8$ $8$ to make its value $1$ $1$

Any number raised to $0$ $0$ is equal to $1$ $1$ . Hence, $x = 0$ $x=0$

$x = 0$ $x=0$

Question 3 of 4

Solve for

x

$x$

x = \log_{6} (\frac{1}{36})

$x=log_6 (1/36)$

$x =$ $x=$ (-2)

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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

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

$\log_{\color{#9a00c7}{6}} \color{#00880a}{\frac{1}{36}}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$\frac{1}{36}$ $1/36$

$a$ $a$	$=$ $=$	$6$ $6$
$x$ $x$	$=$ $=$	$x$ $x$

Substitute the components into the exponent form

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

$\color{#00880a}{\frac{1}{36}}$	$=$ $=$	${\color{#9a00c7}{6}}^x$

Make sure that only

x

$x$ is on the left side

$\frac{1}{36}$ $1/36$	$=$ $=$	$6^{x}$ $6^x$

$\frac{1}{6^{2}}$ $1/(6^2)$	$=$ $=$	$6^{x}$ $6^x$	$36 = 6^{2}$ $36=6^2$

$6^{- 2}$ $6^(-2)$	$=$ $=$	$6^{x}$ $6^x$	Reciprocate $\frac{1}{6^{2}}$ $1/(6^2)$
$- 2$ $-2$	$=$ $=$	$x$ $x$	Equate the exponents since the bases are equal
$x$ $x$	$=$ $=$	$- 2$ $-2$

x = - 2

$x=-2$

Question 4 of 4

Solve for

x

$x$

x = \log_{2} (\frac{1}{64})

$x=log_2 (1/64)$

$x =$ $x=$ (-6)

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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

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

$\log_{\color{#9a00c7}{2}} \color{#00880a}{\frac{1}{64}}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$\frac{1}{64}$ $1/64$

$a$ $a$	$=$ $=$	$2$ $2$
$x$ $x$	$=$ $=$	$x$ $x$

Substitute the components into the exponent form

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

$\color{#00880a}{\frac{1}{64}}$	$=$ $=$	${\color{#9a00c7}{2}}^x$

Make sure that only

x

$x$ is on the left side

$\frac{1}{64}$ $1/64$	$=$ $=$	$2^{x}$ $2^x$

$\frac{1}{2^{6}}$ $1/(2^6)$	$=$ $=$	$2^{x}$ $2^x$	$64 = 2^{6}$ $64=2^6$

$2^{- 6}$ $2^(-6)$	$=$ $=$	$2^{x}$ $2^x$	Reciprocate $\frac{1}{2^{6}}$ $1/(2^6)$
$- 6$ $-6$	$=$ $=$	$x$ $x$	Equate the exponents since the bases are equal
$x$ $x$	$=$ $=$	$- 6$ $-6$

x = - 6

$x=-6$

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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

Fantastic!

Exponent Form

Logarithmic Form

4. Question

Hint

Nice Job!

Exponent Form

Logarithmic Form

Quizzes

$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$	$=$ $=$	$x$
$\log_{\color{#9a00c7}{4}} \color{#00880a}{64}$	$=$ $=$	$x$

$N$ $N$	$=$ $=$	$64$ $64$
$a$ $a$	$=$ $=$	$4$ $4$
$x$ $x$	$=$ $=$	$x$ $x$

$64$ $64$	$=$ $=$	$4^{x}$ $4^x$
$4^{3}$ $4^3$	$=$ $=$	$4^{x}$ $4^x$	$64 = 4^{3}$ $64=4^3$
$3$ $3$	$=$ $=$	$x$ $x$	Equate the exponents since the bases are equal
$x$ $x$	$=$ $=$	$3$ $3$

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

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

$N$ $N$	$=$ $=$	$1$ $1$

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