(a)

(b)

(c)

(d)

Find the accumulated value of a CD of $20,000 for 3 yearsat an interest rate of 3.1% if the money is compounded continuously?

50,690218,760,38421,94921,860

Warm Up: Show YOUR work!

Warm Up

The exponential equation 13.49 .967 1 predicts the number of O-rings

that are expected to fail at the temperature x F on the space shuttles. The O-rings were used to seal the connections between d

x

o

f x

ifferent sections of the shuttleengines. Use a calculator to find the number expected to fail at the temperature of 40 degrees.

Warn Up

.8

The population of a small polynesian island can be modeled by

the equation f(x)=8e . Round off your answers to the nearest integer.a. How many people originally went to this small island? (x=0)b. H

x

ow many people will be living on this island 10 years laterdue to general population growth with no new immigration.

Section 5.2/5.4Exponential and Logarithmic Functionsp. 339: 28-36 (even), 37p. 358: 12-20 (even), 31-38

Definition of the Exponential Function The exponential function f with

base b is defined by f(x) =bx or y = bx Where “b” is a positive

constant other than 1 and greater than 0.

YES f(x) = 2x

f(x) = 10x

f(x) = 3x+1 f(x) = (1/2)x-1

NO f(x) = x2

f(x) = 1x f(x) = (-1)x f(x) = xx

Exponential Growth vs. Decay: y = bx

Growth b>1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,1) As x increases, for b >

1, f(x) also increases without bound

The x-axis (y = 0) is the asymptote

Decay 0<b<1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,1) As x increases, for

0<b<1, f(x) decreases, approaching zero

The x-axis (y = 0) is the asymptote

Example 1Graph the following two equations: f(x)= , f(x)= 44

Draw the asymptotes.

xx

Consider the function y = 2x. How would the graph change given the transformations below?

f(x) = 2x-1

f(x) = 2x + 4

f(x) = 2x - 2

f(x) = 2x + 3

Transformations of Exponential Functions

Transformation

Equations Description

Vertical Translation

G(x) = bx + cG(x) = bx – c

• Shifts the graph of f(x) = bx UP c units

• Shifts the graph of f(x) = bx DOWN c units

Horizontal Translation

G(x) = bx+c

G(x) = bx-c• Shifts the graph of f(x) = bx LEFT

c units• Shifts the graph of f(x) = bx RIGHT

c unitsReflection G(x) = -bx

G(x) = b-x• Reflects the graph f(x) bx about

the x-axis• Reflects the graph f(x) bx about

the y-axisVertical Shrinking or Stretching

G(x) = cbx • If c > 1, vertical stretch• If 0< c < 1, vertical shrink

Horizontal stretching or shrinking

G(x) = bcx • If c > 1, horizontal shrink• If 0< c < 1, horizontal stretch

Example Use the graph of f(x)=4 to obtain the graph of g(x)=4 3.What is the domain and range of each function?

x x

Example 2Use the graph of f(x)=4 to obtain the graph of g(x)=4Find the domain and range for the g(x) function.

x x

Example Use the graph of f(x)=4 to obtain the graph of g(x)=2 4Find the domain and range for the g(x) function.

x x

Logarithmic functions

Definition: Let b > 0 and b not equal to 1. Then y is the logarithm of x to the base be written: y = logbx if and only if by = x

In other words, a logarithmic graph is the inverse of an exponential.

Graphing logarithms To graph y = logb x Rewrite as an

exponential equation: by = x

Make an x/y table, filling in y first.

Graph points. y = log3x

x Y

-1

0

1

2

x

y

Other logarithms

Common Logarithm: (base 10)log x = y 10y = x

Natural Logarithm: (base e)logex = y ln e = y ey = x

Properties of y = logbx y = log bx OR

x = by

Domain

Range

Asymptotes (line that graph

approaches, but does not touch)

Point on all graphs

Determining the Domains of Logarithmic Functions.FYI: the range never changes!

Remember the “argument” must be positive (> 0)

f(x) = log2 (x – 1)

f(x) = (log3 x) – 1

f(x) = log4 |x|

f(x) = log5 ( x2 – 4)

Transformations of Logarithmic Functions

Transformation

Equations Description

Vertical Translation

G(x) = logbx + c

G(x) = logbx - c

• Shifts the graph of f(x)= logbx UP c units

• Shifts the graph of f(x)= logbx DOWN c units

Horizontal Translation

G(x) =logb (x + c)

G(x) = logb (x - c)

• Shifts the graph of f(x)= logbx LEFT c units, VERTICAL ASYMPTOTE (x = -c)

• Shifts the graph of f(x)= logbx RIGHT c units, VERTICAL ASYMPTOTE (x = c)

Reflection G(x) = -logbx

G(x) = logb (-x)

• Reflects the graph f(x)= logbx about the x-axis

• Reflects the graph f(x)= logbx about the y-axis

Vertical Shrinking or Stretching

G(x) =c logbx • If c > 1, vertical stretch• If 0< c < 1, vertical shrink

Horizontal stretching or

shrinkingG(x) = logb(cx)

• If c > 1, horizontal shrink• If 0< c < 1, horizontal stretch

Example3

3

Use transformations to graph g(x)=2+log ( 3). Start with log .

xx

x

y

Example

3Use transformations to graph g(x)=-log x

x

y

Example

3Use transformations to graph g(x)=log ( )x

x

y

Closure

4Find the domain of f(x) = log ( 5)x

0

0

Use the formula R=log to solve the problem. If an

earthquake is 100 times as intense as a zero-level quake(I=100 I ), what is its magnitude on the Richter Scale?

II

2

Find the domain of each function.a. f(x)= ln (x-3)

b. h(x)=ln x