Q A function when the base(a) is some positive number. Q The exponent is variable(x). Q The...

A function when the base(a) is some positive number.

The exponent is variable(x).

The exponential function with base a is defined by:

20. Exponential Functions

Example 1x -2 -1 0 1 2

f(x)xxf 2)(

-1 1 2

2

4

x

f (x)

Domain:

Range:

Horizontal Asymptote:

x -2 -1 0 1 2

f(x)x

xf

3

1)(

-1 1 2

3

9

x

f (x)

Domain:

Range:

Horizontal Asymptote:

Example 2

Find the domain, range, and horizontal asymptote.

35)( xxf

x

f(x) Domain:

Range:

Horizontal Asymptote:

Example 3

Find the domain, range, and horizontal asymptote.

14)( xxf

x

f(x)

Domain:

Range:

Horizontal Asymptote:

Example 4

Special base, e 2.7182818……..

xexf )(

Use a calculator to evaluate the following values of the natural exponential function (round to 5 decimal places):

Natural Base, e

nt

n

rPtA

1)(

An investment has its interest compounded n times a year. The amount the investment is worth in t years is given by:

where:P = r = n = t =

Compound Interest

What would a $5000 investment be worth in 3 years if the interest rate is 7.5% and the investment is compounded:

nt

n

rPtA

1)(

yearly

semiannually

monthly

continuously

n A(3)

Example 5

Exponential functions f (x) = ax are one-to-one functions.

This means they each have an inverse function.

We denote the inverse function with loga, the logarithmic function with base a.

21. Logarithmic Functions

xxf alog)(

Definition

xayx ya log

logax is

Switch from logarithmic form to exponential form:

29log3

11.log10

3

12log8

xayx ya log

Switch from exponential form to logarithmic form:

12553

?49log7

2

116 4

1

?4log16

Evaluating logarithms

x

y

xxf 2log)(

Create a table of points:

x

1/2

1

2

4

1

6

-1 1

-6

xy 2log

Graph

x

y

xxf alog)(

New domain restriction:1. No negative under an even root2. No division by zero3. Only

1

Domain:

Range:

Vertical Asymptote:

Graph

1. loga1 = 0 (you must raise a to the power of 0 in order to get a 1)

2. logaa = 1 (you must raise a to the power of 1 to get an a)

3. logaax = x (you must raise a to the power of x to get ax)

4. alogax = x (logax is the power to which a must be raised to get x)

Properties

xxxf loglog)( 10

With calculator: 5 log)5( f02 log)20( f

Common Logarithm (Base 10)

Without calculator: 001 log)100( f

1. log)1(. f

xxxf e lnlog)(

With calculator: 5 ln)5( f02 ln)20( f

Without calculator: eef ln)( 33 ln)( eef

Natural Logarithm (Base e)

Let A, B, and C be any real numbers with A > 0 and B > 0.

22. Laws of Logarithms

1. loga(AB) = loga A + loga B

2. loga(A/B) = loga A - loga B

3. loga(AC) = C loga A

Applying the laws

Use the laws of logarithms to expand the following:

)(log 3 2 yx

6log5

x

w

xy2

log

Use the laws of logarithms to combine the following:

5 ln3ln

2log54log 33

yx log2

1log3

Applying the laws - continued

To evaluate other bases on the calculator, use the following formula:

a

b

log

logloga b

15log4

2log7

a

b

ln

ln

Change of Base

Isolate exponential function and apply logarithm function to both sides of the equation.

Isolate the logarithm function and apply the base to both sides of the equation.

Remember logarithm laws and inverse properties:

23. Solving equations

Example 1

777:Solve x

1525:Solve 34 xe

Example 2

023:Solve 2 xx ee

Example 3

0)7ln(2:Solve x

Example 4

12)log(x1)log(x:Solve

Example 5

Growth: n(t) = n0ert, positive power (for population models)

Decay: m(t) = m0e-rt, negative power (for decay models)

Cooling: T(t)=Ts+D0e -rt, negative power (indicates loss in difference between object and surrounding temperature)

Logarithms: pH scale, earthquake intensity, decibel levels

24. Exponential Applications

(a) What is the initial number of bacteria?

(b) What is the relative rate of growth? Express your answer as a percentage.

(c) How many bacteria are in the culture after 5 hours? Please round the answer to the nearest integer.

(d) When will the number of bacteria reach 10,000? Please round the answer to the nearest hundredth.

Example 1

.49.

400e )( is speciescertain a of population Thet

tn

.yearper 4% isgrowth of rate observed theand

2005,in 116,000 city wascertain ain population The

(a) Find a function that models the population t years after 2005?

(b) Find the projected population in the year 2016? Please round the answer to the nearest thousand.

(c) In what year will the population reach 200000?

Example 2

.00495

40 )(by given is 210polonium of

sample g40 a from daysafter t remaining )( mass The

t-.etm-

-tm

(a) How much remains after 60 days?

(b) When will 10 grams remain? Please round the answer to the nearest day.

Example 3

Half-life for radioactive isotopes

In general:

(c) Find the half-life of polonium-210.

m(t) = m0e-rt

How long ago was the mummy buried?Round answer to nearest ten.(Carbon-14 has a half-life of 5730 years.)

rtemtm 0 )(Example 4

The burial cloth of an Egyptian mummy is estimated to have 56% of the carbon-14 it contained originally.