Chapter 8 – Exponential and Logarithmic Functions

18 Days

8.1 Modeling Exponential Functions

Five Days

What equations did you come up with to model your data in the penny lab?

Penny Lab

Exponential Functions

decay. lexponentia have we1,b0 If

growth lexponentia have we1,b If

.100 , where

form theoffunction a isFunction lExponentiaAn

, b, bax

bay x

Family of Exponential Functions

Identify the following as either exponential growth, decay, or undefined.

Growth or Decay

x

x

x

x

y

y

y

y

)4(3

)10(5.

)25(.1300

2

Graphing Exponential Functions

x

x

y

y

)4(5.

)25(.100

Lower Dauphin HS had a student population of 1150 kids in the school year ending in 2010. Find the function that models the population if the rate of growth continues at a rate of 2.3% per year.

Modeling Exponential Growth

Find the equation of the exponential function that passes through (4,8) and (6,32).

Writing equations given two points.

Questions??

pg 434 (# 1 - 19 odd)

Homework

pg 434 (32, 42 - 45 all)

Homework

Practice with Exponential Growth and Decay

Homework

8.2 Properties of Exponential Functions

Four Days

Exponential functions are those that have

An example of an exponential function is

What is an exponential function?

power variablebaseconstant

ky

),(

),(

:Asymptote Horizontal

0 :Range

:Domain 14

12

10

8

6

4

2

-2

-15 -10 -5 5 10 15

f x = 2x

kbaf(x) x-h

xxf 2)(

Asymptote Horizontal theis

point.locator theis )1,( where)(

ky

kahkbaxf hx

We can shift exponential function using the same patterns from before.

Shifting Exponential Graphs

Asymptote Horizontal theis

point.locator theis )1,( where)(

ky

kahkbaxf hx

xxf 24)(

224)( xxf

324)( 2 xxf

The half life of a radioactive substance in the time it takes for half of the substance to decay.

Half Life

units) life half as same thebemust (units Time

substance of life Half

substance ofAmount Initial 0

21

0

1

x

h

A

Ay xh

Caffeine has a half life of 5.7 hours in the human body. If you drink a Coca-Cola at noon, how much caffeine will be in your body when you go to bed?

What do we need to know?

What time do you go to bed?

32mg of caffeine per 12oz can.

Half Life Example

The exponential function with base e are very useful in describing continuous growth and decay.

The Natural Function; e

71828.2

1lim 1

e

e x

xx

The Natural Exponential Function

ns.applicatio and smathematic advancedin

functions usefulmost theof one isfunction lexponentia natural The

number x. realevery for

)(

by definded is Function lExponentia Natural Thexexf

f

What is the difference between simple interest and compound interest?

What is continuously compounded interest?

Continuously Compounded Interest

years. ofnumber theis

decimal) a (as rateinterest annual theis

invested principal theis

years after amount total theis

: where

:FormulaInterest Compoundly Continuous The

t

r

P

tA

PeA rt

An initial investment of $35000 is continuously compounded at 8.5% interest. How much is the investment worth after 5 years? After 15 years?

Continuously Compounded Interest

pg 442 (# 2-8 even, 16, 17, 24-26, 36, 37)

Homework

Exponential Growth and Decay WS

Homework

Simple and Compound Interest WS

Homework

8.1 - 8.2 Review

Homework

8.2 Solving Basic Exponential Equations

One Day

We know that in exponential functions the exponent is a variable. When we wish to solve for that variable we have two approaches we can take.

One approach is to use a logarithm. We will learn about these in a later lesson.

The second is to make use of a property called the Equality Property for Exponential Functions.

The Equality Property of Exponential Functions

Basically, this states that if the bases are the same, then we can simply set the exponents equal.

This property is quite useful when we are trying to solve equations involving exponential functions.

Let’s try a few examples to see how it works.

The Equality Property for Exponential Functions

21 xifonly and if Then

.1 and 0 that Suppose21 xbb

bbxx

32x 5 3x 3(Since the bases are the same wesimply set the exponents equal.)

2x 5 x 3x 5 3

x 8

Here is another example for you to try:

Example 1a:

23x 1 21

3x 5

Example 1

How can we solve an equation when the bases are not the same??

32x 3 27x 1

Does anyone have an idea howwe might approach this?

32x 3 27x 1

32x 3 33(x 1) (our bases are now the sameso simply set the exponents equal)2x 3 3(x 1)

2x 3 3x 3

x 3 3

x 6

x 6

Let’s try another one of these.

Example 2

32

116 1 x

Example 3

Example 4122 84 xx

Example 51331 2733 xxx

Solving Exponential Equations WS

Homework

8.3 LogarithmsTwo Days

The logarithm to the base b exponential is defined as:

What is a Logarithm?

xyby bx log

Re-writing in logarithmic form

32=9

xa+b=9

364log4

rGM log

Evaluate

Evaluating Logsx16log8

Evaluate

Evaluating Logsx27log9

The common logarithm is a log that uses base 10. It can be written in either of the two forms:

The Common Logarithm

yy logor log10

Evaluate the following logs:

Evaluating Logs with the Calculator

3

4log

45log

20log

10log

Page 450 (#6-25 all, 41-48 all, 53-61 all)

Homework

8-3 WS (# 1 - 49 odd)

Homework

8.4 Properties of Logarithms

One Day

Properties of Logarithms

Write each logarithmic expression as a single logarithm

zyx

zyx

x

logloglog3

logloglog3

2loglog

2log5log

Expand each logarithm

42

2

2

3

25

log

log

log

2log

zy

wx

z

xy

yx

x

We can use the properties of logs to re-write a logarithmic expression as a single log so that we can evaluate, or solve, the logarithm.

Ex:

Why use properties of logs?

5log21log 5521

Evaluate

Your turn…

4log2log5 22

Questions??

pg 457 (# 11 - 30 all, 33-41 odd)

Homework

8.5 Solving using the Equality Property

Four Days

Property of Equality for Logs

nmnm bb loglog

Solving Logs

xx log2log27log

Solving Logs

8-4 WS (# 14 - 40 even); 7-5 WS (# 11 - 21 odd)

Homework

8.5 Solving Exponential and Logarithmic Equations

Four Days

Guidelines:◦ When solving an exponential equation, you must

isolate the exponential (or write as one exponential on each side) before you take the log of both sides.

◦ When solving a logarithmic equation, you must isolate the logarithm (or write as a single log on each side of the equation) before you raise both sides using a base b.

Solving Logarithmic and Exponential Equations

Logs as Inverse Operations

xxb

xxb

x

xb

b

xlnlog

x

e

and

eln log

:isThat another. one of inverses are and log xb b

Our calculators can only handle logs that are either base 10, or base e. So, we need a way to re-write any log so that it can be entered into the calculator.

The Change of Base Formula:

Change of Base Formula

b

x

b

xxb log

logor

ln

lnlog

Solving Exponentials

32 x

Solving Exponentials

14425 12 x

Solving Exponentials

1062 x

Solving Logarithms

12log x

Solving Logarithms

213log x

Solving Logarithms

94.2log6loglog3 x

Solving Exponentials231 25 xx

Pg 464 #2-44 even, SKIP 20

Homework

Logs that result in Quadratics

Pg 466 #79-96 all, 100-103 all

Homework

Exponential and Logs Review

Homework

8.6 Natural LogarithmsFour Days

7-8 WS Applications of Exp and Log Functions

Homework

7-8 WS Applications of Exp and Log Functions

Homework