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

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Euler s number
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
Definition

  • The base e, which is approximately

    e = 2.718281828…

    is an irrational number called the natural base.


Example 1 page 511
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 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
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
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
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


Definition1
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
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
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
Properties of Natural Logarithmic Functions Logarithmic Function.

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 Logarithmic Function.

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 Logarithmic Function.

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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