CC Exponential and Logarithmic Functions

Section 0.2 Exponential and Logarithmic Functions

In this chapter we define exponential and logarithmic functions. For a more extensive treatment of exponential functions we refer the reader to PreCalculus at Nebraska: Exponential Functions

⁶

mathbooks.unl.edu/PreCalculus/Exponential-Functions.html

and for a more extensive treatment of exponential functions we refer the reader to PreCalculus at Nebraska: Logarithmic Functions

⁷

mathbooks.unl.edu/PreCalculus/section-33.html

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Subsection 0.2.1 Exponential Functions

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Before we discuss exponential functions, it is first useful to define the notion of concavity.

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Concavity.

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The graph of a function is concave up if it bends upward as we move left to right; it is concave down if it bends downward. A line is neither concave up nor concave down.

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An exponential function will be either be increasing or decreasing, but will always be concave up! We define such a function as follows:

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Exponential Function.

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P (t) = a \cdot b^{t}, where b > 0 and b \neq 1, a \neq 0.

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The constant

a

is the

y

-value of the

y

-intercept of the function.

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The equation

P (t) = P_{0} a^{t}

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gives an exponential function with base

a .

Then

\frac{P (t + 1)}{P (t)} = \frac{P_{0} a^{t + 1}}{P_{0} a^{t}} = a = constant rate ofgrowth/decay

Growth: $a > 1 :$ Doubling time: the time it takes to double the initial amount
Decay $0 < a < 1 :$ Half-life: the time it takes to decay to half of the initial amount

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Doubling Time and Half-life.

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The doubling time of an exponentially increasing quantity is the time required for the quantity to double.
The half-life of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half.

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We will (naturally) consider exponentials (and later, logarithms) in base

e \approx 2.718281828459 . . . (irrational number)

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For example, the general exponential function in the natural base

P (t) = P_{0} a^{t} = P_{0} (e^{k})^{t} = P_{0} e^{k t}, a = e^{k} .

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Thus, we will have

exponential growth if $a > 1,$ which gives $k > 0$
exponential decay if $0 < a < 1,$ which gives $k < 0$

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The number

k

is called the continuous rate of growth/decay. To find

k

we will need the logarithmic function.

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Subsection 0.2.2 Logarithmic Functions

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Recall inverse functions. If for each

y

in the range of

f

there exists exactly one value of

x

such that

f (x) = y,

then

f

has an inverse at

y

denoted by

f^{- 1}

such that

f (x) = y ⟺ f^{- 1} (y) = x

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Hence

f (f^{- 1} (y)) = y and f^{- 1} (f (x)) = x .

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The inverse of the exponential function

f (x) = e^{x}

is the natural logarithmic function

f^{- 1} (x) = \ln (x)

so we have

\ln (x) = c ⟺ e^{c} = x

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The Logarithm.

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Let

b \neq 1

be a positive number, then the function

f (t) = \log_{b} (t)

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is called a logarithm with base

b .

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Upon inputting a value

t,

the function

\log_{b} (t)

will tell you the power of

b

which will yield

t .

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Due to the relationship between logarithms and exponentials, we often say that the equations

x = \log_{b} (y) and b^{x} = y

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are equivalent.

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Warning 0.29.

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Two common bases of a logarithm are base

10

and

e .

Since they are used so often, we have developed a short hand notation for a logarithm of base

10

and a logarithm of base

e .

This short hand is shown below:

\log_{10} (y) = \log (y) .

\log_{e} (y) = \ln (y) .

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In other words, we simply drop the subscript when referring to base

10

and we change to

\ln

when referring to base

e

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There are some important properties of logarithms that you should be familiar with.

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Properties of Logarithms.

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x, y, b > 0,

and

b \neq 1,

then

$\log_{b} (x y) = \log_{b} (x) + \log_{b} (y),$
$\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y),$
$\log_{b} (x^{k}) = k \cdot \log_{b} (x),$
$\log_{b} (b^{y}) = y,$
$b^{\log_{b} (x)} = x .$

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Subsection 0.2.3 Supplemental Videos

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Introduction
⁸
unl.yuja.com/V/Video?v=7114207&node=34303316&a=13030261&autoplay=1
Half Life
⁹
unl.yuja.com/V/Video?v=7114205&node=34303257&a=52467459&autoplay=1
Logarithm
¹⁰
unl.yuja.com/V/Video?v=7114203&node=34303274&a=166161527&autoplay=1
Examples
¹¹
unl.yuja.com/V/Video?v=7114209&node=34303222&a=102937779&autoplay=1

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Exercises 0.2.4 Exercises

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1. General Exponential Functions.

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If you write the function

P = 8 e^{2 t}

in the form

P = P_{0} a^{t},

then

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P_{0} =

, and

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a =

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This function represents exponential

growth
decay
neither growth nor decay

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2. Finding Exponential Functions.

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PTX:ERROR: WeBWorK problem local/Functions/Exponential_Functions/PointsOnFunction.pg with seed 8 is either empty or failed to compile Use -a to halt with returned content

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3. Finding the Parameters for an Exponential Function.

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PTX:ERROR: WeBWorK problem local/Functions/Exponential_Functions/FindInitValueAndFactor.pg with seed 9 is either empty or failed to compile Use -a to halt with returned content

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4. Half-Life.

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Find the half-life (in hours) of a radioactive substance that is reduced by

5

percent in

65

hours.

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Half life = (include units

¹²

/webwork2_files/helpFiles/Units.html

)

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5. Applied Half-Life.

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In the year 2005, a picture supposedly painted by a famous artist some time after 1595 but before 1645 contains 96 percent of its carbon-14 (half-life 5730 years).

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From this information, could this picture have been painted by this artist?

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Approximately how old is the painting? years

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6. Solving Exponential Equations.

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PTX:ERROR: WeBWorK problem local/Functions/Logarithms/Solving.pg with seed 12 is either empty or failed to compile Use -a to halt with returned content

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7. Using Inverse Functions With Exponentials and Logarithms.

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PTX:ERROR: WeBWorK problem local/Functions/Logarithms/Inverses.pg with seed 13 is either empty or failed to compile Use -a to halt with returned content

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