4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read...

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4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents? What is the definition? How do you rewrite log equations? What are two special log values? What is a common log? A natural

Transcript of 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read...

Page 1: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

4.4 Evaluate Logarithms & Graph Logarithmic Functions

p. 251What is a logarithm? How do you read

it? What relationship exists between logs and exponents? What is the definition?

How do you rewrite log equations?What are two special log values?

What is a common log? A natural log?What logs can you evaluate using a

calculator?

Page 2: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Evaluating Log Expressions

• We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6?• Because 22<6<23 you would expect

the answer to be between 2 & 3.• To answer this question exactly,

mathematicians defined logarithms.

Page 3: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Definition of Logarithm to base b

• Let b & x be positive numbers & b ≠ 1.• The logarithm of x with base a is

denoted by logbx and is defined:

•logbx = y if by = x• This expression is read “log base b of x”

• The function f(x) = logbx is the logarithmic function with base b.

Page 4: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

• The definition tells you that the equations logbx = y and by = x are equivalent.

• Rewriting forms:

• To evaluate log3 9 = x ask yourself…

• “Self… 3 to what power is 9?”

• 32 = 9 so…… log39 = 2

Page 5: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Log form Exp. form

•log216 = 4

•log1010 = 1

•log31 = 0

•log10 .1 = -1

•log2 6 ≈ 2.585

•24 = 16•101 = 10•30 = 1•10-1 = .1•22.585 = 6

Page 6: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Evaluate

•log381 =

•Log5125 =

•Log4256 =

•Log2(1/32) =

•3x = 81•5x = 125•4x = 256•2x = (1/32)

4

34

-5

Page 7: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Evaluating logarithms now you try some!

•Log 4 16 = •Log 5 1 =•Log 4 2 =•Log 3 (-1) =• (Think of the graph of y=3x)

20

½ (because 41/2 = 2) undefined

Page 8: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

You should learn the following general forms!!!

•Log b 1 = 0 because b0 = 1

•Log b b = 1 because b1 = b

•Log b bx = x because bx = bx

Page 9: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Natural logarithms

•log e x = ln x

•ln means log base e

e

Page 10: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Common logarithms

•log 10 x = log x

•Understood base 10 if nothing is there.

Page 11: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Common logs and natural logs with a calculator

log10 button

ln button**Only common log and natural log bases are on a calculator.

Page 12: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Keystrokes

Expression Keystrokes Display

a. log 8

b. ln 0.3

Check

8

.3

0.903089987

–1.203972804

100.903 8

0.3e –1.204

Page 13: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

TornadoesThe wind speed s (in miles per hour) near the center of a tornado can be modeled by:

where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado’s center.

93 log d + 65s =

Page 14: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Solution

= 93 log 220 + 65Write function.

93(2.342) + 65

= 282.806

Substitute 220 for d.

Use a calculator.

Simplify.

The wind speed near the tornado’s center was about 283 miles per hour.

ANSWER

93 log d + 65s =

Page 15: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

• What is a logarithm? How do you read it? A logarithm is another way of expressing an

exponent. It is read log base b of y. • What relationship exists between logs and

exponents? What is the definition?• logax = y if ay = x• How do you rewrite logs?The base with the exponent on the other side of

the = .• What are two special log values?Logb1=0 and logbb=1• What is a common log? A natural log?Common log is base 10. Natural log is base e.• What logs can you evaluate using a

calculator?Base 10

Page 16: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

4.4 AssignmentPage 255, 3-6, 8-16,

20-26 even

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4.4 Day 2

• How do you use inverse properties with logarithms?

• How do you graph logs?

Page 18: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

•g(x) = log b x is the inverse of

•f(x) = bx

•f(g(x)) = x and g(f(x)) = x•Exponential and log functions

are inverses and “undo” each other.

Page 19: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

•So: g(f(x)) = logbbx = x• f(g(x)) = blog

bx = x

•10log2 = •Log39x =•10logx =•Log5125x =

2Log3(32)x =Log332x=2x

x3x

Page 20: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Use Inverse Properties

Simplify the expression.

a. 10log4 b. 5

log 25x

SOLUTION

Express 25 as a power with base 5.

a. 10log4 = 4

b. 5

log 25x = (52 ) x5

log

=5

log 52x

2x=

Power of a power property

blog xb = x

blog bx = x

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Find Inverse PropertiesFind the inverse of the function.

SOLUTION

b.

a. y = 6 x b. y = ln (x + 3)

a.

6log

From the definition of logarithm, the inverse ofy = 6x is y= x.

Write original function.y = ln (x + 3)Switch x and y.x = ln (y + 3)Write in exponential form.Solve for y.

=ex (y + 3)=ex – 3 y

ANSWER The inverse of y = ln (x + 3) is y = ex – 3.

Page 22: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Use Inverse PropertiesSimplify the expression.

SOLUTION

10. 8 8log x

8 8log x = x

blog bb = x

11. 7

log 7–3x

SOLUTION

7log 7–3x = –3x

alog ax = x

Exponent form

Log form

Log form

Exponent form

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Use Inverse PropertiesFind the inverse of14. y = 4x

SOLUTION

From the definition of logarithm, the inverse of

4log y = x.is

y = ln (x – 5).Find the inverse of15.

y = ln (x – 5)

SOLUTION

Write original function.Switch x and y.x = ln (y – 5)Write in exponential form.Solve for y.

=ex (y – 5)=ex + 5 y

ANSWER The inverse of y = ln (x – 5) is y = e x + 5.

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Finding Inverses

• Find the inverse of:

•y = log3x• By definition of logarithm, the inverse

is y=3x • OR write it in exponential form and

switch the x & y! 3y = x 3x = y

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Finding Inverses cont.

• Find the inverse of :

•Y = ln (x +1)•X = ln (y + 1) Switch the x &

y

•ex = y + 1 Write in exp form

•ex – 1 = y solve for y

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4.4 Graphing Logs

p. 254

Page 27: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Graphs of logs

•y = logb(x-h)+k •Has vertical asymptote x=h•The domain is x>h, the range

is all reals•If b>1, the graph moves up to

the right•If 0<b<1, the graph moves

down to the right

Page 28: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

Graphing a log functionGraph the function.

SOLUTION

a. y = 3log x

Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote.

From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.

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Graph y = log1/3x-1

• Plot (1/3,0) & (3,-2)

• Vert line x=0 is asy.

• Connect the dots

X=0

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Graph y =log5(x+2)

• Plot easy points (-1,0) & (3,1)

• Label the asymptote x=−2

• Connect the dots using the asymptote.

X=-2

Page 31: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

• How do you use inverse properties with logarithms?

Exponential and log functions are inverses and “undo” each other.

• How do you graph logs?

Pick 1, the base number, and a power of the base for x.

Page 32: 4.4 Evaluate Logarithms & Graph Logarithmic Functions p. 251 What is a logarithm? How do you read it? What relationship exists between logs and exponents?

4.4 Assignment Day 2

• Page 256, 28-44 even, 45-51 odd