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

The Exponential and Logarithmic Functions

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The Exponential and Logarithmic Functions

Natural

Section 6.3

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

- The base e, which is approximately
e = 2.718281828…

is an irrational number called the natural base.

Use your calculator to evaluate the following.

Round our answers to 4 decimal places.

7.3891

0.1353

1.3499

103.0455

The function f, represented by

f(x) = cex

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

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, ∞).

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

- 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

- 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.

- 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.

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

y=exacross the liney= x.

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 )