The Exponential and Logarithmic Functions

Section 6.3

Natural

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

To eleven decimal places,

e = 2.71828182846

Value of e

The base e, which is approximately e = 2.718281828…

is an irrational number called the natural base.

Definition

Use your calculator to evaluate the following. Round our answers to 4 decimal places.

7.3891 0.1353

1.3499

103.0455

Example 1 – Page 511

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 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

f(x) = cex

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

•1 unit right, down 3 units•h.a. y = -3•y-int: f(0) = -2.6•Domain: (-∞, ∞)•Range: (-3, ∞).

• reflect x-axis•h.a. y = 0•y-int: f(0) = -1•Domain: (-∞, ∞)•Range: (-∞, 0).

• reflect y-axis, down 5•h.a. y = -5•y-int: f(0) = -4•Domain: (-∞, ∞)•Range: (-5, ∞).

Example 3 – Page 514

Natural Exponential Growth and Decay

The function of the form P(t) = P0ekt Models exponential growth if k > 0 and exponential decay when k < 0.

T = timeP0 = 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

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 4 – Page 516

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.

a) Find and interpret V(0) and V(2).

b) What is the Jeep’s value in 2007?

Example (Problem 57– Page 526

What is the Natural Logarithmic Function?What is the Natural Logarithmic Function?

• Logarithmic Functions with Base 10 are called “common logs.”

• 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 = ln x.

y = ln x if and only if x=ey.

Definition

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

Evaluate the following.

Example 7 – Page 518

Graphs: Natural Exponential Function and Natural Logarithmic Function.The graph of y = lnx is a reflection of the graph of y =ex across the line y = x.

Properties of Natural Logarithmic Functions

• Domain: (0, ∞)

• Range: (-∞, ∞)

• x-intercept is (1,0)

• Vertical asymptote x = 0.

• f is 1-1 (one-to-one) 

f(x) = ln x

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 2V.A. x = 4Domain: (4, ∞)Range: (-∞, ∞)

Reflect x axis down 2V.A. x = 0Domain: (0, ∞)Range: (-∞, ∞)

Example 8 – Page 519

Find the domain of each function algebraically.

(31, ∞ )

(-∞, 2.7 )

f(x) = ln (x-31)

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

Example 9 – Page 521