Chapter 10 Exponential and Logarithmic Functions

Chapter 10

Exponential and Logarithmic Functions

Lesson 10.1:Exponential Functions

Learning Targets:

• I can graph exponential functions.

• I can determine if an exponential function is growth or decay.

• I can write an exponential function given two points.

• I can solve equations involving exponents.

Graphing an exponential function:

y = a∙bx standard form of an exponential function

a = y-intercept (0, a)

b = base

x = exponent

Example 1: Graphing Exponential FunctionsSketch the graph of y = 4x and identify its domain and range.

Domain:________

Range:_________

Example 2: Graphing Exponential FunctionsSketch the graph of y = 0.7x and identify its domain and range.

Domain:________

Range:_________

What type of function is it?

Growth

Decay

Example 3: Indicate whether each exponential function is growth or decay.

xy )7.0(xy )2(

3

1

x

y

5

210

DecayGrowthDecay

Example 4: Write an exponential function whose graph passes through the given points.

(0, -2) and (3, -54)y = a∙bx


(0, 7) and (1, 1.4) y = a∙bx


(0, 3) and (-1, 6) y = a∙bx


(0, -18) and (-2, -2)

y = a∙bx

Remember the Exponent Rules:

5 3 5 2

6 5

6

Example 8: Simplify expressions.

a.

b.

Example 8: Simplify expressions.

2 5 2 3

7 3

7

c. d

.

Example 9: Solve equations.

2564 29 nStep 1: Make the bases the same.Step 2: Set the exponents equal.

Step 3: Solve.


125 93 xx


322 13 x

Home Fun10-1 Worksheet

Lesson 10-2: Logarithmic Functions

Objectives: I can….

Convert from logarithmic to exponential

form and vice versa.

Evaluate logarithmic expressions.

Solve logarithmic equations.

Definition of Logarithm:

Let b > 0 and b 1. Then n is the logarithm of m to the

base b, written

logb m = n if and only if bn = m

Check it out!Exponential FormLogarithmic Form

meansmeansmeansmeansmeans

means

means

416log2

01log2

24log2

12log2

12

1log2

38log2

24

1log2

1624

823

422

221

120

2

12 1

4

12 2

Example 1: Convert to exponential form.

29log3 2log 1001

10 a.

b.

Example 1 (continued)

281log9

log319 2

c.

d.

Flower Power Root Rule

mn bb nm

PowerFlower

Root RootRoot

Power

Example 2: Convert to logarithmic

form.

53 125

2713 3a

.b.


34 81

8112 9c

.d.

Example 3: Evaluate logarithmic expressions.

log3 243a.


log10 1000b.

A couple of intricate ones…

log9 92

7log7 (x2 1)

c.

d.

Think-Pair-Share!

log5 53

3log3 (x2)a.

b.

Example 4: Solve logarithmic

equations.

log8 n 43

a.


log27 n 23b

.

Example 4 (still continued!)

log4 x2 log4(4x 3)c.

Example 4 (last one!)

log5 x2 log5(x 6)d.

Home FunHome Fun10-2 10-2 WorksheetWorksheet

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Lesson 10-3: Properties of Logs

Learning Targets:

I can use the product and quotient properties of logs.

I can use the power property of logs.

I can solve equations using properties of logs.

Properties

yxy

xbbb logloglog

xnx bn

b loglog

Product Property:

Quotient Property:

Power Property:

yxyx bbb logloglog Example: 10log2loglog 777 x

Example:

Example:

15log2loglog 222 y

64loglog2 77 x

Example 1: Solving EquationsExample 1: Solving Equations

125log5loglog4 222 x

Example 2: Solving EquationsExample 2: Solving Equations

2)12(loglog 88 xx

Your Turn : Solve each equation.

a.

b.

Your Turn : Solve each equation.

a.

b.4)6(loglog 22 xx

24log6log2log2 333 x

Home

10-3 Worksheet


Algebra 2A – Lesson 10-4

Common Logarithms

Lesson 10-4: Common Logs

Learning Targets:

I can find common logarithms.

I can solve logarithmic and exponential equations.

I can use the Change of Base Formula.

What is a Common Logarithm?

m10log

This logarithm is used so frequently, that it is programmed into our calculators.

We write it as: log m Note that we don’t write the

base for a common log.

Example 1: Find Common Logs with a Calculator

Example 1: Find Common Logs with a Calculator

Use a calculator to evaluate each logarithm to four decimal places.

a) log 6 b) log 0.35No base

labeled, so it must be log10 (the common

log).

Change of Base Formula

b

aab

10

10

log

loglog

This is a useful formula, because now we can rewrite ANY log as log10

Example 2: Use the Change of Base Formula

Express each log in terms of common logs.

Then, approximate its value to four decimal places.

a) log3 16 b) log2 50

Example 3: Use logs to solve equations where the power is the variable. If necessary, round to four decimal places.

a) Solve: 5x

= 62

You Try: 3x

= 17

If necessary, round to four decimal places.

b) Solve: 72x+1

= 11


You Try: 64x-3

= 8


Home Fun

10-4 Worksheet

Algebra 2B - Algebra 2B - Chapter 10Chapter 10Section 5Section 5

Natural LogarithmsNatural Logarithms

Lesson 10-6: Exponential Growth and Decay Story Problems

Learning Targets:

I can solve problems involving exponential growth (with doubling)

I can solve problems involving exponential decay (with half-life)

Growth and Decay Problemsxaby

a = initial amount of something

b (the growth factor) is written as )1( r

r = the growth or decay rate

x = time (as given in the problem)

b > 1 indicates a growth problem

0 < b < 1 indicates a decay problem

GROWTH DECAY

xray )1( xray )1(

xay )2(x

ay

2

1

Doubling Half-life

Doubling

xray )1( xray )1(

Half-life

Growth Decay

Example 1: DoublingAn experiment begins with 300 bacteria and the

population doubles every hour. How many bacteria will there be after:

a) 2 hours?

b) 10.5 hours?

Example 2: Decay Problem

Suppose a car you bought new for $35,000 in 2008 depreciates at a rate of 18% per year.

a. Write an equation for the car’s value x years after 2008.

b. What will the car’s value be after 5 years?

Example 3: GrowthA computer engineer is hired for a salary of $70,400. If she gets a 5% raise each year, after how many years will she be making $100,000 or more?

Example 4: Half-lifeRadium-226 has a half-life of 1,620 years.

a) Write an equation for the percent of Radium-226 remaining if there is currently 550 grams after x half-life periods.if there currently 550 grams after x half-life periods.

b) If you begin with 4 grams of Radium-226, how much will remain after three half-life periods?

c) How many years are equal to three half-life periods of Radium-226?

Practice Story Problem 1The population of a certain strain of bacteria grows according to the formula y = a(2)x, where x is the time in hours.

If there are now 50 bacteria, how many will there be in 2 days?

The population N of a certain bacteria grows according to

the equation N = 200(2)1.4t, where t is the time in hours.

a) How many bacteria were there at the beginning of the

experiment?

b) In how many hours will the number of bacteria reach 100,000?

Practice Story Problem 2

In 2001, the population of Lagos, Nigeria was about 7,998,000. Use the population growth rate of 4.06% per year

a. Estimate the population in 2009.

b. In about how many years will the population be over 50,000,000?


You bought a car for $28,500 in 2014. It depreciates at 13% each year?


a. What is the value of the car in 2018?

b. In how many years will the car depreciate to $5000?

An isotope of Cesium-137 has a half-life of 30 years.

a. If you start with 20 mg of the substance, how many mg will be left after 90 years?

b. After 120 years?



In 2010, the population of Australia was 17,800,000. In 2014, the population is now 22,000,000. At what rate is the population growing?

Home practice

10-6 Worksheet

Algebra 2B

Lesson 10.5:

Natural Logarithms

Learning Targets:

I can understand and use base e.

I can solve base e equations and write equivalent expressions.

Base e

• “e” is used extensively in finance and business

• Euler’s number: e

n

n

11• As n increases,

approaches the value e ≈ 2.71828.

Base e and Natural Log

• The functions y = ex and y = ln x are inverse functions.

• A couple interesting properties:xe x ln xex ln

• Find the e key and the LN key on your calculators.

Example 1: Write Equivalent Equations Write an equivalent

logarithmic or exponential equation.a) ex = 23 b) ln x ≈

1.2528

Example 1: Write Equivalent Equations Write an equivalent

logarithmic or exponential equation. c) ex = 6 d) ln x = 2.25

Example 2: Evaluate Natural Logarithms

a) eln 21 12

ln xeb)

Example 3: Solve Equations

a) 1043 2 xe

Example 3: Solve Equations

b) 1552 2 xe

Pert Formula

rtPeA

Example 4: Solve Pert ProblemsSuppose you deposit $700 into an account paying 6% annual interest, compounded continuously.

a) What is the balance after 8 years?

b) How long will it take for the balance in your

account to reach at least $2000?

Your Turn: Pert ProblemsSuppose you deposit $1100 into an account paying 5.5% annual interest, compounded continuously.

a) What is the balance after 8 years?

b) How long will it take for the balance in your account to reach at least $2000?

Home Practice10-5 Worksheet


ClosureSolve.

24 g of a substance has a half-life of 18 years. How much of the substance will remain after 72 years?

Documents

Chapter 10 Exponential and Logarithmic Functions