Int-Alg Chapter 10 Logarithmic Functions

Section A.10 Chapter 10 Logarithmic Functions

Subsection A.10.1 Logarithmic Functions

Subsubsection A.10.1.1 Graph log functions

One way to graph a log function is to first make a table of values for its inverse function, the exponential function with the same base, then interchange the variables.

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Checkpoint A.10.1.

Complete the table of values and graph on the same grid:

f (x) = x^{3}

and

g (x) = 3^{x}

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$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$8$	$10$
$f (x)$
$g (x)$

$Grid for 3 to x and x-cubed$

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Use your tables from part (a) to graph $h (x) = \sqrt[3]{x}$ and $j (x) = \log_{3} (x)$ on the same grid.
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$Grid for log base 3 and cube root$

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Answer.

$3 to x and x-cubed$

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$log base 3 and cube root$

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Subsubsection A.10.1.2 Use function notation

A log function is the inverse of the exponential function with the same base, and vice versa.

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Example A.10.2.

For each function

f (x),

decide whether

f (a + b) = f (a) + f (b) .

$f (x) = 3^{x}$

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$f (x) = \log_{3} (x)$

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Solution.

$f (a + b) = 3^{a + b},$ and $f (a) + f (b) = 3^{a} + 3^{b} .$
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But $3^{a + b}$ is not equivalent to $3^{a} + 3^{b};$ in fact $3^{a + b} = 3^{a} \cdot 3^{b} .$
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So for this function, $f (a + b) \neq f (a) + f (b) .$
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$f (a + b) = \log_{3} (a + b),$ and $f (a) + f (b) = \log_{3} (a) + \log_{3} (b) .$
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But $\log_{3} (a + b)$ is not equivalent to $\log_{3} (a) + \log_{3} 9;$ in fact $\log_{3} (a b) = \log_{3} (a) + \log_{3} (b) .$
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So for this function, $f (a + b) \neq f (a) + f (b) .$
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Checkpoint A.10.3.

g (x) = 5^{x} .

Evaluate and simplify if possible.

$g (3 + t)$

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$g (3 t)$

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Answer.

$125 \cdot 5^{t}$

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$125^{t}$

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Example A.10.4.

q (x) = 9^{x}

and

p (x)

is its inverse function. Evaluate if possible.

$q (\frac{1}{2})$

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$p (3)$

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$q (0)$

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$p (0)$

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Solution.

$3$

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$\frac{1}{2}$

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$1$

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undefined

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Checkpoint A.10.5.

h (x) = \log_{4} (x) .

Evaluate if possible.

$h (4)$

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$g (4),$ where $g$ is the inverse function for $h$

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$h (0)$

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$g (0)$

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Answer.

$1$

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$256$

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undefined

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$1$

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Checkpoint A.10.6.

f (x) = \log_{8} (x) .

Evaluate and simplify if possible.

$f (64 p)$

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$f (64 + p)$

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Answer.

$2 + \log_{8} (p)$

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cannot be simplified

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Subsubsection A.10.1.3 Use the properties of logarithms

The three properties of logarithms are helpful in making computations involving logs.

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Properties of Logarithms.

x,

y,

b > 0,

and

b \neq 1,

then

$\log_{b} (x y) = \log_{b} (x) + \log_{b} (y)$
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$\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$
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$\log_{b} (x^{k}) = k \log_{b} (x)$
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Example A.10.7.

\log_{b} (10) = 2.303

and

\log_{b} (2) = 0.693,

what is

\log_{b} (5) ?

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Solution.

Because

5 = \frac{10}{2},

\log_{b} (5) = \log_{b} (\frac{10}{2}) = \log_{b} (10) - \log_{b} (2) = 2.303 - 0.693 = 1.61

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Example A.10.8.

\log_{b} (10) = 2.303

and

\log_{b} (2) = 0.693,

what is

\log_{b} (20) ?

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Solution.

Because

20 = 10 \cdot 2,

\log_{b} (20) = \log_{b} (10 \cdot 2) = \log_{b} (10) + \log_{b} (2) = 2.303 + 0.693 = 2.996

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Checkpoint A.10.9.

Take the log of each number. What do you notice?

$8 \cdot 100 = 800$

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$12 \cdot 1000 = 12, 000$

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$20 \cdot 25 = 500$

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$200 \cdot 250 = 50, 000$

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Answer.

$\log (8) + \log (10) = \log (800)$

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$\log (12) + \log (100) = \log (12, 000)$

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$\log (20) + \log (25) = \log (500)$

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$\log (200) + \log (250) = \log (50, 000)$

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Checkpoint A.10.10.

Compare the two operations. What do you notice?

(i) Compute $10^{2.68}$ (ii) Solve for $x : \log (x) = 2.68$

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(i) Compute $10^{- 0.75}$ (ii) Solve for $x : \log (x) = - 0.75$

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Answer.

(i) and (ii) have the same answer: $478.63$

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(i) and (ii) have the same answer: $0.1778$

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Checkpoint A.10.11.

The ratio of $N$ to $P$ is $32.6 .$ Compute $\log (N) - \log (P) .$

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$\log (z) - \log (t) = 2.5 .$ Compute $\frac{z}{t} .$

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Answer.

$1.5132$

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$316.2278$

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Subsection A.10.2 Logarithmic Scales

Subsubsection A.10.2.1 Plot a log scale

Because

\log (x)

grows very slowly, we can use logs to compare quantities that vary greatly in magnitude.

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Example A.10.12.

Complete the table. Round the values to one decimal place.
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$x$ $1$ $5$ $25$ $125$ $625$

$\log (x)$

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Plot the values of $x$ on a log scale.
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Each time we multiply $x$ by 5, how much does the logarithm increase? What is $\log (5),$ to one decimal place?

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Solution.

$x$ $1$ $5$ $25$ $125$ $625$

$\log x$ $0$ $0.7$ $1.4$ $2.1$ $2.8$

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Each time we multiply $x$ by 5, the log of $x$ increases by 0.7, because $\log (5) = 0.7 .$ This is an application of the log properties:
$\log (5 x) = \log (x) + \log (5) = \log (x) + 0.7$

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Checkpoint A.10.13.

Complete the table. Round the values to one decimal place.
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$x$ $5$ $10$ $20$ $40$ $80$

$\log (x)$

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Plot the values of $x$ on a log scale.
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Each time we multiply $x$ by 2, how much does the logarithm increase? What is $\log (2),$ to one decimal place?

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Answer.

$x$ $5$ $10$ $20$ $40$ $80$

$\log (x)$ $0.7$ $1$ $1.3$ $1.6$ $1.9$

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$0.3; 0.3$

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Checkpoint A.10.14.

Complete the table. Round the values to one decimal place.
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$x$ $0.25$ $1$ $4$ $16$ $64$ $256$

$\log (x)$

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Plot the values of $x$ on a log scale.
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Each time we multiply $x$ by 4, how much does the logarithm increase? What is $\log (4),$ to one decimal place?

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