3.2 logarithmic functions 2015 digital lesson. 3.1 warm-up mr. smith deposited $6,500 in an account...

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3.2 Logarithmic Functions 2015 Digital Lesson

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Page 1: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

3.2Logarithmic Functions

2015

Digital Lesson

Page 2: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

3.1 Warm-upMr. Smith deposited $6,500 in an account that pays

the account pays 4.5% interest, compounded monthly.

How much will he have in the account after 6 years?

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Page 3: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

HWQ 9/8Mr. Jones received a bonus equivalent to

15% of his yearly salary and has decided to deposit

it in a savings account in which interest is compounded

continuously. His salary is $68,500 per year and

the account pays 5.5% interest.

How much interest will his deposit earn after 5 years?

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Page 4: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

What is a log?

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2log 8 because

Page 5: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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For x 0 and 0 a 1, y = loga x if and only if x = a y.

The function given by f (x) = loga x is called the

logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

Exponential function: y = ax

Logarithmic function: y = logax is equivalent to x = ay

A logarithm is an exponent!

Page 6: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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y = log2( )21 = 2

y

2

1

Examples: Write the equivalent exponential equation and solve for y.

1 = 5 yy = log51

16 = 4y y = log416

16 = 2yy = log216

SolutionEquivalent Exponential

Equation

Logarithmic Equation

16 = 24 y = 4

2

1= 2-1 y = –1

16 = 42 y = 2

1 = 50 y = 0

Page 7: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

Converting between Forms

Log Form Exponential Form

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Page 8: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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log10 –4 LOG –4 ENTER ERROR

no power of 10 gives a negative number

The base 10 logarithm function f (x) = log10 x is called the

common logarithm function.

The LOG key on a calculator is used to obtain common logarithms.

Examples: Calculate the values using a calculator.

log10 100

log10 5

Function Value Keystrokes Display

LOG 100 ENTER 2

LOG 5 ENTER 0.69897005

2log10( ) – 0.3979400LOG ( 2 5 ) ENTER

Page 9: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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Examples: Solve for x: log6 6 = x log6 6 = 1 property 2 x = 1

Simplify: log3 35

log3 35 = 5 property 3

Simplify: 7log7

9

7log7

9 = 9 property 3

Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a.

4. If loga x = loga y, then x = y. one-to-one property 3. loga a

x = x and alogax = x inverse property

Page 10: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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x

y

Graph f (x) = log2 x

Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

83

42

21

10

–1

–2

2xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

38

24

12

01

–2

log2 xx1

41

2–1

Page 11: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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Example: Graph the common logarithm function f(x) = log10 x.

by calculator

1

10

10.6020.3010–1–2f(x) = log10 x

10421x 1

100

y

x5

–5

f(x) = log10 x

x = 0 vertical

asymptote

(0, 1) x-intercept

Page 12: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)

3. x-intercept (1, 0)

5. increasing6. continuous7. one-to-one 8. reflection of y = a

x in y = x

1. domain ),0( 2. range ),(

4. vertical asymptote )(0 a s 0 xfxx

Graph of f (x) = loga x (a 1)

x

y y = x

y = log2 x

y = a x

domain

range

y-axisvertical

asymptote

x-intercept(1, 0)

Page 13: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3ln –2ln 100

Function Value Keystrokes Display

LN 3 ENTER 1.0986122ERRORLN –2 ENTER

LN 100 ENTER 4.6051701

y = ln x

(x 0, e 2.718281)

y

x5

–5

y = ln x is equivalent to e y = x

Page 14: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

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Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression.

2

1ln

e 2ln 2 e inverse property

2 0l ne 2 0 inverse property

eln3 3)1(3 property 2

00 1ln property 1

Page 15: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

Finding the Domains of Logarithmic Functions

Find the Domain of each Function:

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) ln 2a f x x ) ln 2b f x x 2, , 2

y

x5

–5

y

x5

–5

Page 16: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.

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6logf x xy

x5

–5

: 0,Domain

: 0VA x

int : 1,0x

Page 17: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.

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10log 1 4g x x y

x5

–5

: 1,Domain

: 1VA x

4int : 1 10 ,0x

Page 18: 3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded

Homework

• Pg. 195 1-19 odd, 25-29 odd, 35-47 odd

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