Chapter 4Exponential and Logarithm Functions

4.1 – Exponential Functions

4.2 – The Natural Exponential Function

4.3 – Logarithm Functions

4.4 – Logarithmic Transformations

4.5 – Logistic Growth

Section 4.1Exponential Functions

Review of Laws of Exponents (p157 & 158)

Characteristics of Exponential Functions

Modeling with Exponential Functions

Basic Exponential Functions

x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

y=4x

f(x) = 4x

domain: reals

range: positive reals

y intercept: 1

increasing

a > 1

Basic Exponential Functions: f(x) = 4x

as x → ∞, 4x → ∞

as x → -∞, 4x → 0

y = 0 is a Horizontal Asymptote

Basic Exponential Functions: f(x) = 4x

as x increases without bound,

4x increases without bound

as x decreases without bound,

4x gets close to 0

x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

y=4x

x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

y=(¼)x

f(x) = (¼)x

domain: reals

range: positive reals

y intercept: 1

decreasing

0 < a < 1

Basic Exponential Functions: f(x) = (¼)x

x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

y=(¼)x

as x → ∞, (¼)x → 0 y = 0 is a Horizontal Asymptote

as x → -∞, (¼)x → ∞

Basic Exponential Functions: f(x) = (¼)x

as x increases without bound,

(¼)x gets close to 0.

as x decreases without bound,

(¼)x increases without bound

f(x) = 4x and g(x) = (¼)x

graphs are reflections about the y axis

Does f(-x) = g(x)?

Basic Exponential Functions

f(x) = 4x

domain: reals

range: positive reals

HA: y = 0

y intercept: 1

f(x) = 5*4x

domain: reals

range: positive reals

HA: y = 0

y intercept: 5

f(x) = (a)x and f(x) = c*ax

Basic Exponential Functions

f(x) = c*ax

domain: reals

range: positive reals

HA: y = 0

y intercept: c

increasing for a > 1decreasing for 0 < a < 1

Basic Exponential Functions: f(x) = c*ax

Characteristics

f(x) = 5*4x

domain: reals

range: positive reals

HA: y = 0

y intercept: 5

f(x) = 20 + 5*4x

domain: reals

range: reals > 20

HA: y = 20

y intercept: 25

Variations of Basic Exponential Functions

Modeling with Exponential Functions

Example/156 A math student pours himself a mug of steaming coffee and then forgets to drink it. In a room that remains at 20C, the coffee cools, losing heat rapidly at first and then more slowly as the liquid approaches room temperature. The coffee is initially 90C and after 10 minutes cools to 68C.

Find a model for the temperature of the coffee over time.

exponential with 0 < a < 1 and shifted up 20 units

H(t) = V + c*at

H(t) = V + c*at

Can we solve for V, c, and a?

H(0) = 9020 + c*a0 = 9020+c = 90c = 70 so H(t) = 20 + 70*at

Modeling with Exponential Functions

V = 20 so H(t) = 20 + c*at

H(10) = 6820 + 70*a10 = 6870*a10 = 48a10 = 48/70a = (48/70)(1/10)

a = 0.963 so H(t) = 20 + 70*(0.963)t

Vertical shift

Initial Temperature

Another data point

Modeling with Exponential Functions

Example/156 A gymnastics team practices its balance-beam routine, improving month by month. Initially the average score is 3.8 but then scores increase rapidly; as more time passes, additional efforts result in small gains. In fact, after 6 months of practice, the average score is 5.7

Find a model for the average score of the team over time.

exponential with 0 < a < 1, reflected about x axis and shifted up.

S(t) = V - c*at

S(t) = V - c*at

Can we solve for V, c, and a?

S(0) = 3.810 - c*a0 = 3.810 - c = 3.8c = 6.2 so S(t) = 10 – 6.2*at

Modeling with Exponential Functions

V = 10 so S(t) = 10 - c*at

S(6) = 5.710 – 6.2*a6 = 5.7-6.2*a6 = -4.3a6 = -4.3/-6.2a = (4.3/6.2)(1/6)

a = 0.94 so S(t) = 10 – 6.2*(0.94)t

Vertical shift

Initial Score

Another data point

Section 4.2The Natural Exponential Function

What is e?

Base-e Exponential Functions

x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

y=ex

f(x) = ex

domain: reals

range: positive reals

y intercept: 1

increasing

a > 1

Natural Exponential Function: f(x) = ex = (2.718)x

Homework

Pages 193-194

#1-16

Turn In: #7,#8, #13