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5.1 Exponential Functions

Rules For Exponents If a 0 and b 0, the

following hold true for all real numbers x and y.

5.1 Exponential Functions

If we apply the quotient rule, we get

5.1 Exponential Functions

- For any nonzero number x

and

5.1 Exponential Functions

- Examples

5.1 Exponential Functions

Examples

- (5x-2)3 125x-6125/x6
- (3x/y3)2 9x2/y6
- (4x)-1 1/(4x)
- (2a3b-3c4)3 8a9b-9c12

- 40 1
- 2-1 ½
- (½)-2 4
- 5-2 1/25

5.1 Exponential Functions

Simplify

5.1 Exponential Functions

5.1 Exponential Functions

Simplify

Rewrite

Notice

5.1 Exponential Functions

Simplify

Rewrite

Notice

5.1 Exponential Functions

Simplify

Rewrite

Notice

5.1 Exponential Functions

- Then

- If

Examples

- Since

- Since

- Since

5.1 Exponential Functions

In general, if n is a multiple of m, then

5.1 Exponential Functions

Use the rules for exponents to solve for x

- 4x 128
- (2)2x 27
- 2x 7
- x 7/2

- 2x 1/32
- 2x 2-5
- x -5

5.1 Exponential Functions

- 27x 9-x1
- (33)x (32)-x1
- 33x 3-2x2
- 3x -2x 2
- 5x 2
- x 2/5

- (x3y2/3)1/2
- x3/2y1/3

5.1 Exponential Functions

Definition Exponential Function Let a be a

positive real number other than 1, the function

f(x) ax is the exponential function with base

a.

5.1 Exponential Functions

y 2 x

- If b 1, then the graph of b x will
- Rise from left to right.
- Not intersect the x-axis.
- Approach the x-axis.
- Have a y-intercept of (0, 1)

5.1 Exponential Functions

y (1/2) x

- If 0
- Fall from left to right.
- Not intersect the x-axis.
- Approach the x-axis.
- Have a y-intercept of (0, 1)

5.1 Exponential Functions

Natural Exponential Function where e is the

natural base and e ? 2.718

5.1 Exponential Functions

(-8, 8)

(-8, 8)

(-8, 8)

(0, 8)

(0, 8)

(0, 8)

Dec.

Inc.

Inc.

(0, 1)

5.1 Exponential Functions

Use translation of functions

to graph the following.

Determine the domain and range f (x) 2(x

2) 3

Domain (-8, 8) Range (-3, 8)

5.1 Exponential Functions

Definitions Exponential Growth and Decay

The function y k ax, k 0 is a model for

exponential growth if a 1, and a model for

exponential decay if 0 y new amount yO original amount b

base t time h half life

5.1 Exponential Functions

- An isotope of sodium, Na, has a half-life of 15

hours. A sample of this isotope has mass 2 g. - Find the amount remaining after t hours.
- Find the amount remaining after 60 hours.

- b. y yobt/h
- y 2 (1/2)(60/15)
- y 2(1/2)4
- y .125 g

- a. y yobt/h
- y 2 (1/2)(t/15)

5.1 Exponential Functions

- A bacteria double every three days. There are 50

bacteria initially present - Find the amount after 2 weeks.
- When will there be 3000 bacteria?

- a. y yobt/h
- y 50 (2)(14/3)
- y 1269 bacteria

5.1 Exponential Functions

A bacteria double every three days. There are 50

bacteria initially present When will

there be 3000 bacteria?

- b. y yobt/h
- 3000 50 (2)(t/3)
- 60 2t/3

5.2 Simple and Compound Interest

Formulas for Simple Interest Suppose P dollars

are invested at a simple interest rate r, where r

is a decimal, then P is called the principal and

P r is the interest received at the end of one

interest period.

5.2 Simple and Compound Interest

Formulas for Compound Interest After t years,

the balance A in an account with principal P and

annual interest rate r is given by the two

formulas below.

1. For n compoundings per year

2. For continuous compounding

5.2 Simple and Compound Interest

Find the balance after 10 years if 1000.00 is

invested at 4 and the account pays simple

interest.

5.2 Simple and Compound Interest

Find the balance after 10 years if 1000.00 is

invested at 4 and the interest is compounded

1485.95

a. Semiannually

b. Monthly

1490.83

c. Continuously

1491.82

5.3 Effective Rate and Annuities

Effective Annual Rate The effective annual rate

of ieff of APR compounded k times per year is

given by the equation Another name for

effective annual rate is effective yield

5.3 Effective Rate and Annuities

What is the better rate of return, 7 compounded

quarterly or 7.2 compounded semianually?

5.3 Effective Rate and Annuities

1.071 1 .071 7.1

1.073 1 .073 7.3

7.2 compounded semiannually is better.

5.3 Effective Rate and Annuities

What is the better rate of return, 8 compounded

monthly or 8.2 compounded quarterly?

5.3 Effective Rate and Annuities

8.3

8.5

8.2 quarterly is better.

5.3 Effective Rate and Annuities

Future Value of an Ordinary Annuity The Future

Value S of an ordinary annuity consisting of n

equal payments of R dollars, each with an

interest rate i per period is

5.3 Effective Rate and Annuities

Suppose 25.00 per month is invested at 8

compounded quarterly. How much will be in the

account after one year?

- 1st quarter 25.00
- 2nd quarter 25.00(1.08/4) 25.00 50.50
- 3rd quarter 50.50(1.08/4) 25.00 76.51
- 4th quarter 76.51(1.08/4) 25.00 103.04

5.3 Effective Rate and Annuities

Present Value of an Ordinary Annuity The Present

Value A of an ordinary annuity consisting of n

equal payments of R dollars, each with an

interest rate i per period is

5.4 Logarithmic Functions

The inverse of an exponential function is called

a logarithmic function.

Definition x a y if and only if y log a x

5.4 Logarithmic Functions

- log 4 16 2 ? 42 16
- log 3 81 4 ? 34 81
- log10 100 2 ? 102 100

5.4 Logarithmic Functions

Sketch a graph of f (x) 2x and sketch a graph

of its inverse. What is the domain and range of

the inverse of f.

Domain (0, 8) Range (-8, 8)

5.4 Logarithmic Functions

The function f (x) log a x is called a

logarithmic function.

- Domain (0, 8)
- Range (-8, 8)
- Asymptote x 0
- Increasing for a 1
- Decreasing for 0
- Common Point (1, 0)

5.4 Logarithmic Functions

Find the inverse of g(x) 3x.

Note The function and its inverse are

symmetrical about the line y x.

5.4 Logarithmic Functions

Find the inverse of g(x) ex.

ln x is called the natural logarithmic function

5.4 Logarithmic Functions

So

So

So

So

5.4 Logarithmic Functions

- loga(ax) x for all x ? ?
- alog ax x for all x 0
- loga(xy) logax logay
- loga(x/y) logax logay
- logaxn n logax

Common Logarithm log 10 x log x Natural

Logarithm log e x ln x All the above

properties hold.

5.4 Logarithmic Functions

Product Rule

5.4 Logarithmic Functions

Quotient Rule

5.4 Logarithmic Functions

Power Rule

5.4 Logarithmic Functions

Expand

5.4 Logarithmic Functions

Find an equation of best fit for the data (1,3),

(2,12), (3,27), (4,48)

5.5 Graphs of Logarithmic Functions

The function f (x) log a x is called a

logarithmic function.

- Domain (0, 8)
- Range (-8, 8)
- Asymptote x 0
- Increasing for a 1
- Decreasing for 0
- Common Point (1, 0)

5.5 Graphs of Logarithmic Functions

The natural and common logarithms can be found on

your calculator. Logarithms of other bases are

not. You need the change of base formula.

where b is any other appropriate base. (usually

base 10 or base e)

5.5 Graphs of Logarithmic Functions

Sketch the graph of

Domain (2,?) Range (-?, ?)

5.5 Graphs of Logarithmic Functions

Sketch the graph of

Domain (-2,?) Range (-?, ?)

5.5 Graphs of Logarithmic Functions

Sketch the graph of

Domain (-3,?) Range (-?, ?)

5.5 Graphs of Logarithmic Functions

On the Richter scale, the magnitude R of an

earthquake can be measured by the intensity

model.

R Magnitude a Amplitude T Period B

Damping Factor

5.5 Graphs of Logarithmic Functions

What is the magnitude on the Richter scale of

an earthquake if a 300, T 30 and B 1.2?

5.6 Solving Exponential Equations

- Solve 4 3x 16 x 2
- The bases can be rewritten as
- (22) 3x (24) (x 2)
- 2 6x 2 4x 8
- 6x 4x 8
- 2x -8
- x -4

5.6 Solving Exponential Equations

- To solve exponential equations, pick a convenient

base (often base 10 or base e) and take the log

of both sides. - Solve

5.6 Solving Exponential Equations

- Take the log of both sides

- Power rule

5.6 Solving Exponential Equations

- Solve for x

- Divide

5.6 Solving Exponential Equations

- To solve logarithmic equations, write both sides

of the equation as a single log with the same

base, then equate the arguments of the log

expressions. - Solve

5.6 Solving Exponential Equations

- Write the left side as a single logarithm

5.6 Solving Exponential Equations

- Equate the arguments

5.6 Solving Exponential Equations

- Solve for x

5.6 Solving Exponential Equations

5.6 Solving Exponential Equations

- Check for extraneous solutions.

5.6 Solving Exponential Equations

- To solve logarithmic equations with one side of

the equation equal to a constant, change the

equation to an exponential equation - Solve

5.6 Solving Exponential Equations

- Write the left side as a single logarithm

5.6 Solving Exponential Equations

- Write as an exponential equations

5.6 Solving Exponential Equations

- Solve for x

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