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The Exponential and Logarithmic Functions. Natural. Section 6.3. Euler’s Number.

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The Exponential and Logarithmic Functions

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The Exponential and Logarithmic Functions

Natural

Section 6.3


Euler’s Number

This value is so important in mathematics thatit has been given its own symbol, e, sometimes called Euler’s number. The number e hasmany of the same characteristics as π. Its decimal expansion never terminates or repeatsin a pattern. It is an irrational number.

Value of e

To eleven decimal places,

e = 2.71828182846


Definition

  • The base e, which is approximately

    e = 2.718281828…

    is an irrational number called the natural base.


Example 1 – Page 511

Use your calculator to evaluate the following.

Round our answers to 4 decimal places.

7.3891

0.1353

1.3499

103.0455


The Natural Exponential Function

The function f, represented by

f(x) = cex

is the natural exponential function, where c is the constant, and x is the exponent.


Properties of Natural Exponential Function

Example

f(x) = cex

Properties of an natural exponential function:

  • Domain: (-∞, ∞)

  • Range: (0, ∞)

  • y-intercept is (0,c)

  • f increases on (-∞, ∞)

  • The negative x-axis is a horizontal asymptote.

  • f is 1-1 (one-to-one) and therefore has an inverse.


Example 3 – Page 514

State the transformation of each function, horizontal asymptote, y-intercept, and domain and range for each function.

  • reflect x-axis

  • h.a. y = 0

  • y-int: f(0) = -1

  • Domain: (-∞, ∞)

  • Range: (-∞, 0).

  • 1 unit right, down 3 units

  • h.a. y = -3

  • y-int: f(0) = -2.6

  • Domain: (-∞, ∞)

  • Range: (-3, ∞).

  • reflect y-axis, down 5

  • h.a. y = -5

  • y-int: f(0) = -4

  • Domain: (-∞, ∞)

  • Range: (-5, ∞).


Natural Exponential Growth and Decay

The function of the form P(t) = P0ektModels exponential growth if k > 0 and exponential decay when k < 0.

T = time

P0 = the initial amount, or value of P at time 0, P > 0

k = is the continuous growth or decay rate

(expressed as a decimal)

ek= growth or decay factor


Example 4 – Page 516

For each natural exponential function, identify the initial value, the continuous growth or decay rate, and the growth or decay factor.

.

  • Initial Value : 100

  • Growth Rate: 2.5%

  • Growth Factor: = 1.0253

  • Initial Value : 500

  • Decay Rate: -7.5%

  • Decay Factor: = 0.9277


Example (Problem 57– Page 526

  • Ricky bought a Jeep Wrangler in 2003. The value of his Jeep can by modeled by V(t)=25499e-0.155t where t is the number of years after 2003.

  • Find and interpret V(0) and V(2).

  • What is the Jeep’s value in 2007?


What is the Natural Logarithmic Function?

  • Logarithmic Functions with Base 10 are called “commonlogs.”

    • log (x) means log10(x) - The Common Logarithmic Function

  • Logarithmic Functions with Base e are called “natural logs.”

    • ln (x) means loge(x) - The Natural Logarithmic Function


Definition

  • Let x > 0. The logarithmic function with base e is defined as y = logex. This function is called the natural logarithm and is denoted by y = lnx.

  • y = lnx if and only if x=ey.


Basic Properties of Natural Logarithms

  • ln (1)

  • ln (e)

  • ln (ex)

ln (1) = loge(1) = 0 since e0= 1

ln(e) = loge(e) = 1 since 1 is the exponent that goes on e to produce e1.

ln (ex) = loge ex = x since ex= ex

= x


Example 7 – Page 518

Evaluate the following.


Graphs: Natural Exponential Function and Natural Logarithmic Function.

The graph of y= lnxis a reflection of the graph of

y=exacross the liney= x.


Properties of Natural Logarithmic Functions

f(x) = ln x

  • Domain: (0, ∞)

  • Range: (-∞, ∞)

  • x-intercept is (1,0)

  • Vertical asymptote x = 0.

  • f is 1-1 (one-to-one)


Example 8 – Page 519

For each function, state the transformations applied to y = lnx. Determine the vertical asymptote, and the domain and range for each function.

b. f(x) = ln(x-4) + 2

c. y = -lnx - 2

4 Right, Shift Up 2

V.A. x = 4

Domain: (4, ∞)

Range: (-∞, ∞)

Reflect x axis down 2

V.A. x = 0

Domain: (0, ∞)

Range: (-∞, ∞)


Example 9 – Page 521

Find the domain of each function algebraically.

f(x) = ln (x-31)

f(x) = ln (5.4 - 2x) + 3.2

(31, ∞ )

(-∞, 2.7 )


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